BOCK
HOLSTEIN
PRODUCTION
m PLANNING
and
CONTROL
UNIVERSITY
OF FLORIDA
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COLLEGE LIBRARY
*% &
PRODUCTION
PLANNING
AND
CONTROL
Digitized by the Internet Archive
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PRODUCTION
PLANNING
AND
CONTROL:
TEXT AND READINGS
ROBERT H. BOCK
School of Business Administration
Northwestern University
WILLIAM K. HOLSTEIN
Krannert Graduate School
of Industrial Administration
Purdue University
CHARLES E. MERRILL BOOKS, INC.
COLUMBUS, OHIO
Copyright © 1963, by Charles E. Merrill Books, Inc., Colum
bus, Ohio. All rights reserved. No part of this book may be
reproduced in any form, by mimeograph or any other means,
without permission in writing from the publisher.
Library of Congress Catalog Card Number: 637389
PRINTED IN THE UNITED STATES OF AMERICA
Preface
Production management is concerned with the function of produc
ing goods or services in largescale industrial activity. It is the counter
part of marketing management, which is concerned with the function
of selling the goods or services. In terms of business organization, the
chief executive charged with fulfilling the production function is usu
ally called the production manager, or manufacturing manager in the
particular case of manufacturing operations. This book is about the
functions of the production manager.
We are further concerned with the analytical underpinnings of pro
duction management. The treatment, although oriented toward these
management science aspects, is not about analysis as such. That is,
we prefer to treat operating problems in an analytical framework
rather than analytical problems with operating examples. We are thus
centered on the functional decisions of the production manager and
have therefore utilized only those analytical tools which have been
sufficiently proven in application.
Mathematics is employed — its use could hardly be avoided in an
analytical presentation. Since we employ only proven analytical tools
such as linear programming and simulation, however, the amount and
level of mathematics is limited. In two areas where the reader's mathe
matics might not suffice— linear programming and calculus — appen
dices are provided to build on the usual background of algebra and
trigonometry. This foundation is ample for the subject matter, and
the book is entirely selfcontained as far as mathematics is concerned.
To analyze the production management function, we have selected
a blend of readings and text. The readings are utilized because in many
rapidly developing analytical areas, an expert can say far better, and
with greater authority, what we could only attempt to rephrase. The
readings also result in the inclusion of more updated material, an
example of which is the treatment of PERT and heuristics in produc
tion planning.
The textual material is designed to complement the readings by in
troducing and tying together the three major parts and surveying the
conventional approaches to the general area. The reader is thereby
provided with a survey of traditional methods and their deficiencies
before he begins his examination of the analytical treatment.
In addition to the introductory material, editorial notes are pro
vided wherever the readings discuss a narrow aspect of the problem
in question. Thus the editorial notes broaden the readings and pre
vent them from leaving the reader with a limited perspective. Finally,
the appendices provide the necessary mathematical base, so that the
readings are comprehensible to the reader with a minimal mathematics
background. We feel that this arrangement combines the strength of
readings (currentness and expertness) with the advantages of text
(continuity and completeness) .
We believe that the direction suggested by Production Planning and
Control: Text and Readings — the analytical approach to production
management — offers one of the really challenging concepts in business
management.
We wish to express our thanks to the publishers and authors whose
works we have drawn upon, and who have cooperated with us in the
selection of illustrations for this book.
Evans ton, Illinois
Lafayette, Indiana
VI
Contents
•PART I — Planning and Controlling
Production Levels
Chapter 1. INTRODUCTION TO PART I, 3
Definitions . . . Conventional methods . . . Gantt Charts
. . . The shortcomings of conventional methods . . . Mod
ern analytical methods . . . Linear decision rules . . . PERT
. . . Heuristics
Chapter 2. PRODUCTION PLANNING WITH LINEAR
PROGRAMMING
I. Do You Want Production or Profit?, 13
NYLES v. reinfeld
II. Applying Linear Programming to the
Plywood Industry, 22
ERNEST KOENIGSBERG
Chapter 3. MATERIALS PLANNING WITH LINEAR
PROGRAMMING
III. A Linear Programming Application to
Cupola Charging, 39
R. W. METZGER AND R. SCHWARZBEK
Chapter 4. PRODUCTION PLANNING WITH THE
TRANSPORTATION MODEL
IV. Mathematical Programming: Better Infor
mation for Better Decision Making, 53
ALEXANDER HENDERSON AND
ROBERT SCHLAIFER
V. Production Scheduling by the Transporta
tion Method of Linear Programming, 102
EDWARD H. BOWMAN
Chapter 5. PRODUCTION CONTROL WITH LINEAR
DECISION RULES
VI. Mathematics for Production Scheduling, 106
M. ANSHEN, C. HOLT, F. MODIGLIANI,
J. MUTH, H. SIMON
vii
VII. Linear Decision Rules and Freight Yard
Operations, 119
EDWIN MANSFIELD AND HAROLD H. WEIN
Chapter 6. THE PERT SCHEDULING TECHNIQUE
VIII. Program Evaluation and Review Technique, 130
DAVID G. BOULANGER
IX. How to Plan and Control with PERT, 140
ROBERT W. MILLER
Chapter 7. HEURISTICS IN PRODUCTION
SCHEDULING
X. A Heuristic Method of Assembly Line Balancing, 159
MAURICE D. KILBRIDGE AND LEON WESTER
•PART II — Inventory Control
Chapters. INTRODUCTION TO PART II, 177
Importance . . . Functions . . . Relationship to production
planning . . . Extensions to analytical treatments
Chapter 9. BASIC FUNCTIONS
XL Guides to Inventory Policy I. Functions and
Lot Sizes, 179
JOHN F. MAGEE
Chapter 10. UNCERTAINTY PROBLEMS IN
INVENTORY CONTROL
XII. Guides to Inventory Policy II. Problems
of Uncertainty, 200
JOHN F. MAGEE
Chapter 11. COMBINED PROBLEMS OF INVENTORY
AND PRODUCTION CONTROL
XIII. Guides to Inventory Policy III. Anticipating
Future Needs, 225
JOHN F. MAGEE
Chapter 12. RELATED LOTSIZE AND DYNAMIC
PROBLEMS
XIV. An Investigation of Some Quantitative Rela
tionships Between BreakEven Point Analysis
and Economic LotSize Theory, 242
WAYLAND P. SMITH
viii
Chapter 13. SIMULATION IN INVENTORY CONTROL
XV. Determining the "Best Possible" Inventory Levels, 258
KALMAN JOSEPH COHEN
. . . Editors' Note on Simulation, 274
Chapter 14. STATISTICAL METHODS IN INVENTORY
CONTROL
XVI. Physical Inventory Using Sampling Methods, 279
MARION R. BRYSON
XVII. Inventory Policy by Control Chart, 290
J. W. DUDLEY
PART III — Facilities Planning
Chapter 15. INTRODUCTION TO PART III, 305
Importance . . . Conventional approaches . . . Payback
analysis . . . Discounting . . . Present value . . . An eco
nomic model . . . Reconciliation of conventional approaches
Chapter 16. UNCERTAINTY PROBLEMS IN
FACILITIES PLANNING
XVIII. Capital Budgeting and Game Theory, 314
EDWARD G. BENNION
. . . Editors' Note on Game Theory, 329
Chapter 17. FACILITIES PLANNING WITH
MATHEMATICAL MODELS
XIX. Mathematical Models in Capital Budgeting, 331
JAMES C. HETRICK
XX. A Model for Scale of Operations, 363
EDWARD H. BOWMAN AND
JOHN B. STEWART
Appendix A: AN INTRODUCTION TO LINEAR
PROGRAMMING, 372
A simple problem . . . Graphical solution . . . A tabular
method . . . The simplex method . . . Summary . . .
A general discussion of linear programming
Appendix B: THE FUNDAMENTALS OF CALCULUS, 392
The Concept of a function . . . Maxima and Minima in
functions of one variable . . . In functions of several vari
ables . . . Constrained optima . . . Integration
Index, 409
ix
PRODUCTION
PLANNING
AND
CONTROL
PART I
Planning and Controlling
Production Levels
Chapter 1
INTRODUCTION TO PART I
The material in this section focuses on the management functions involved
in the production process — planning, scheduling, and control.
Production planning and production scheduling are often considered as
two names for the same activity. This idea is erroneous, but understand
able, since in many ways the two functions are quite similar. Both produc
tion planning and scheduling set the levels at which the production process
will operate in the future; and both assign responsibilities for accomplishing
the production job. The major differences between planning and scheduling
lie in the time span covered by production plans and the amount of detail
in the plan.
Production planning involves setting production levels for several periods
in the future and assigning general responsibility to provide data for making
decisions on the size and composition of the labor force, capital equipment
and plant additions, and planned inventory levels. The ability to meet
demand levels generated by possible alternative sales programs is also a
function of production planning.
At this point the phrase "setting production levels for several periods in
the future" may need some clarification. Production plans are used for many
different purposes. One example is the use of a production plan to help
determine the amount of new capital equipment to be purchased in the
future. In this instance a plan covering the next five, eight, or even ten
years would be required and would indicate the production job to be done
and the capital equipment necessary to accomplish this job.
At the same time that a production plan covering the next several years
is necessary, another plan covering a much shorter time period might also
be called for. This plan might cover only the next few months and might
be used to set aggregate production rates to meet forecasted demand and
planned future inventory levels.
4 PLANNING AND CONTROLLING PRODUCTION LEVELS
Thus, at any given time a company may require several production
plans, each covering a different time period and each used for a different
purpose.
Production scheduling typically covers a much shorter time period than
production planning. Production schedules determine how production re
quired in the next several days or weeks will be assigned to specific depart
ments, processes, machines, and operators in order to meet real deadlines
imposed by the sales department and desired inventory levels.
Whereas most production planning is concerned only with aggregate
productive facilities such as "the packaging department," a production
schedule must stipulate orders in more detail, using such units as "packaging
line #1" or "Warner and Swasey Lathe #6." In addition, in the strictest
definition of the term, a production schedule should stipulate whether Tillie
or Mary pushes the appropriate buttons on the appropriate machine. In
actual practice, this decision is usually left to the foreman to make on the
spot.
Production control involves the constant readjustment of plans and
schedules in the light of collected operating facts. Any production plan, and
most production schedules, are based on some forecast of future demand,
and the only certain element in any forecast is error. As new forecasts are
made to account for recent sales and inventory positions, and apparent
changes in future trends, production plans and schedules must be updated.
Production control is somewhat analogous to maintaining the proper
idling speed on an automobile engine. The car owner is faced with two
related decisions — how often to adjust the idling speed of his automobile,
and, once he has decided to make an adjustment, how great an adjustment
to make. A sports car enthusiast who tinkers with his car every Saturday
morning need make only very minor adjustments on his carburetor since
the car has had only one week to get out of adjustment. A more typical
driver would readjust the carburetor only once a year, but at that time
would make a fairly sizable adjustment.
And so it goes with production control. A company can constantly revise
and update forecasts and make numerous adjustments in production plans,
or it can make more sizable adjustments at less frequent intervals.
It may also be advisable for a company to take little or no action even in
the light of significant differences between actual and forecasted demands.
This would be the case when the variation between actual and forecasted
demand could be attributed purely to random factors and not to any trend
or longterm deviation from forecasted figures. A strategy like this one
would be justifiable if the cost of changing the production level was large
relative to the cost of carrying the extra inventory to protect against devia
tion from forecasts.
INTRODUCTION TO PART I 5
With these broad definitions and differences in mind, we are now in a
position to look at some of the tools which have been developed to cope
with problems in planning, scheduling, and control.
CONVENTIONAL METHODS
Schematic drawings and graphs have proved very useful in the areas of
production planning and scheduling. A simple graph can, for example, bring
into focus the differences between alternate production plans for meeting
forecasted demand requirements. To illustrate one technique, assume that
a company faces the following demand forecast for the coming year:
Forecasted
Cumulative
Production
Cumulative
Demand
Demand
Days
Production
Month
(Units)
(Units)
in Month
Days
January
1,000
1,000
22
22
February
1,000
2,000
20
42
March
1,500
3,500
22
64
April
2,500
6,000
21
85
May
4,000
10,000
23
108
June
4,000
14,000
21
129
July
5,500
19,500
22
151
August
7,500
27,000
23
174
September
10,500
37,500
20
194
October
11,000
48,500
23
217
November
9,000
57,500
22
239
December
2,000
59,500
21
260
By plotting the forecasted production requirements (assuming that we want
to meet expected demand), the extreme seasonal variation becomes very
apparent (Figure 1).
Two alternative production plans might be considered for meeting these
production requirements. Plan I could stipulate a constant or uniform pro
duction rate to be used without variation throughout the year. Plan II could
follow demand exactly, changing the production rate whenever demand
changes. The difference between these extreme plans is indicated by
plotting the cumulative production requirements versus time, where time
now represents production days rather than months. (Production rates must
be expressed in "units per day" rather than "units per month," since not all
months have the same number of days.)
From the diagram in Figure 2 it is obvious that Plan I gives rise to very
sizable inventories throughout the year, which, of course, are costly. In this
simplified example, Plan II requires no inventory, since as soon as the units
are completed, they are sold. Perhaps not so obvious, but nonetheless im
portant, is the magnitude of the required production rates shown by this
diagram. Plan I calls for production at the rate of 23 1 units per day through
6 PLANNING AND CONTROLLING PRODUCTION LEVELS
out the year; whereas the production rate under Plan II varies from month
to month. The Plan II production rate is 45 units/day in January, 174
units/day in June, 525 units/day in September, and 95 units/day in
December. Since the peak production rate under Plan II occurs during the
month of September, the production facility must have a capacity of 525
units/day if Plan II is to be satisfied. This capacity is more than twice that
required by Plan I and, like inventory, productive capacity is expensive.
11000
10000 •
9000
7000 ■
4000
3000
1000
Jan. Feb. Mar. Apr. May June July Aug. Sept. Oct. Nov. Dec.
MONTH
Figure 1
Specific costs attached to carrying inventory, size of plant, and changing
production rates would enable us to determine an optimal production plan
which, no doubt, would lie somewhere between the two extremes discussed
here. Considerations other than costs, such as company policy toward the
labor force, the labormanagement agreement, desired customer service,
and community relations, might also have considerable effect on the pro
duction planning operation. 1
1 The problem of the effect of uncertainty in production and inventory control is
examined in more detail in Chapter 10, "Uncertainty Problems in Inventory Control."
INTRODUCTION TO PART I 7
The most popular schematic scheduling and control technique currently
in use is the Gantt Chart. There are many forms of the basic Gantt Chart
available, each with specific variations on the main theme to suit particular
needs. The Productrol Board, Remington Rand's SchedUGraph, and the
Boardmaster are three popular varieties.
100 150 200
Cumulative Production Days
250
300
Figure 2
There are two basic types of Gantt Charts — the load chart and the plan
ning chart. The most important contribution of the load chart is in keeping
track of previous schedules and available machines; while the planning or
progress chart is used mainly as a control device to plan and measure actual
performance against the plan.
In the Gantt Load Chart, light lines indicate work actually scheduled.
In Figure 3, for example, Machine A is fully loaded for all of week #1.
Machine B, on the other hand, has slightly less than a 50 percent load. The
heavy bars indicate cumulative production capacity for the period covered
by the chart. The cumulative load lines are often used to balance loads
8 PLANNING AND CONTROLLING PRODUCTION LEVELS
among machines and in conjunction with maintenance programs.
WEEK 1
WEEK 2
WEEK 3
WEEK 4
WEEK 5
WEEK 6
MACHINE A
MACHINE B
"
MACHINE C
MACHINE D
Figure 3. Gantt Load Chart
Y
MONDAY
TUESDAY
WEDNESDAY
THURSDAY
FRIDAY
SATURDAY
MACHINE A
#07291
#07386
"I T #07274
..
#06315
MACHINE B
•—
'04118 i r
#05994
#0
b112
MACHINE C
~ #0573
7 ~j
#05593

MACHINE D
#06123
y
.
05382 "I l
tanSd b
#06143 ~
#07914
r "
Figure 4. Gantt Progress Chart
In the Gantt Progress Chart (Figure 4), the lines have similar meanings.
The light lines indicate jobs scheduled; the brackets denote the scheduled
INTRODUCTION TO PART I 9
beginning and end of each job, the number of which is given above the line.
The heavy line indicates the fraction of the job completed at a given time.
The present date is noted with a V at the top of the chart.
Figure 4 shows Machine A working well ahead of schedule. Machine B
is also ahead of schedule. Machine C, which should be half finished with
job #05593, is almost a day behind. Machine D is right on schedule. This
chart focuses attention on the delay at Machine C and should suggest action
to expedite job #05593.
Charts, and in fact many graphical techniques, are often difficult to
handle. The Gantt Chart, for example, requires constant attention to keep
the information timely. Corrections and changes in schedules and plans are
difficult to make, and, as a result, Gantt Charts are often rather messy.
Another weakness of Gantt Charts and similar planning devices is that
they handle only one dimension — time. Quite often there are other im
portant factors which should be considered. For example, in deciding
whether to assign a specific job to Machine A or Machine B, the most im
portant consideration (when there is time available on both machines) is
the relative cost of producing on one machine rather than the other. At the
present time, graphical techniques do not take cost considerations into
account.
MODERN ANALYTICAL METHODS
The past several years have evidenced the development of a rigorous, yet
practical technique which solves the time problem in planning and schedul
ing, and also considers cost. The method, called linear programming, has
proven an extremely useful addition in this area.
Developed during the Second World War for the Air Force, linear pro
gramming has been widely accepted and applied to a broad variety of prob
lems. As a planning and scheduling technique, it assigns production to
machines or production centers in a manner that yields leastcost production
plans. The linear programming model takes into account the fact that
different machines have different efficiencies, require different amounts of
maintenance and operator time, and therefore have different operating
costs. As a planning technique, linear programming not only allows an
optimal allocation of products to machines, but also aids in deciding
which of many possible products to produce.
An introduction to the elements of linear programming is presented in
Appendix A. Basic linear programming can be mastered without an ex
tensive knowledge of mathematics. It is this reason, perhaps, that has made
linear programming so popular. Appendix A concentrates on building a
10 PLANNING AND CONTROLLING PRODUCTION LEVELS
maximum facility for handling linear programming problems while using a
minimum of mathematics. Students interested in a more rigorous and com
plete treatment of linear programming fundamentals are referred to the
many excellent texts currently available on this subject. 2 Having mastered
Appendix A, the reader is equipped to study the application of linear pro
gramming methods to production planning problems in Chapters 2 and 3.
It has been found that some linear programming problems can be solved
by a relatively simple technique known as the transportation method. The
name springs from the fact that the first problems solved by this method
involved transportation or distribution plans. Chapter 4 presents an excel
lent description of the transportation method of linear programming and
several realistic examples of the types of problems that can be handled by
this method. This Chapter also demonstrates the significant advantages in
scheduling by programming rather than by chart or graph.
Granted that mathematical programming represents a significant im
provement over schematic and graphical analysis, there still remain many
unsolved problems — especially in the area of production control. Linear
programming is of little value when planning broad production strategy
for the future and adapting production plans to sales forecast errors. Such
problems become quite involved, and the element of uncertainty disrupts
an orderly analysis by programming techniques.
To cope with large, complex production problems, a faculty group from
the Carnegie Institute of Technology developed a mathematical technique
to adapt plans to changing conditions. The end result of this research was
the formulation of decision rules for setting and controlling both production
and inventory levels and the size of the work force. Chapter 5 describes the
result of this research and presents an application. 3
Another recent development has proven successful in control and sched
uling problems. The development is usually called PERT for Programming
and Evaluation Review Technique. The Air Force uses a version called
PEP. Civilian applications, until recently found mostly in the construction
industry and in research and development planning, use the term "Critical
Path Scheduling." "Arrow Diagramming" is a term also used in connection
with this technique.
2 Three texts on basic linear programming are A. Charnes,W. W. Cooper, A. Hen
derson, An Introduction To Linear Programming, John Wiley & Sons, Inc., 1953; S. I.
Gass, Linear Programming Methods and Applications, McGrawHill Book Co., Inc.,
1959', and Anmin Chung, Linear Programming, Columbus, Ohio: Charles E. Merrill
Books, Inc., 1963.
m
3 The methodology is analyzed more thoroughly in C. C. Holt, Franco Modigliani,
J. F. Muth, H. A. Simon, Planning Production, Inventories and Work Force, Prentice
Hall, Inc., 1960. See especially Chapters 2, 3, and 4.
INTRODUCTION TO PART I
11
To understand PERT more fully, consider the information shown in
Figure 5.
BODY PARTS
ORDERED
4
BODY
ASSEMBLED
\ 4
8 /
5
AUTO FINISHED
ENGINE PARTS
ORDERED
ENGINE
ASSEMBLED
Figure 5. Arrow Diagram
This diagram shows, in oversimplified form, the steps necessary to pro
duce an automobile. Constraints on the sequence of steps are shown (i.e.,
the engine cannot be assembled before the engine parts are ordered; the
body cannot be assembled before the body parts are ordered; and the
finished automobile results from assembled engines and bodies). An esti
mate of the required time is given on the arrow connecting the steps. The
arrow diagram is basic to all PERT applications.
Two paths lead to the finished automobile. The top path requires no less
than eight days: assuming that the body parts are ordered at a zero point
in time, the parts will be available four days later, and assembly will require
an additional four days. The total time required by the top path, therefore,
is no less than eight days.
The bottom path takes at least thirteen days, five for parts to arrive and
eight for assembly. The bottom path is, therefore, the critical path in terms
of slack time or cushion. Progress on the top path could be delayed up to
five days and the automobile would still be produced in thirteen days. How
ever, any delay in the critical path would mean that the automobile would
not be produced in that amount of time.
With a knowledge of the critical path, planners can be made aware of
possible delays and can expedite progress by using resources from slack
paths. In our example, manpower could be transferred from body assembly
to engine assembly. Engine assembly might be cut to six days, and body as
sembly increased to six days. In this way, two days could be cut from the
12 PLANNING AND CONTROLLING PRODUCTION LEVELS
final date for the finished automobile. The revised top path would be 4 f 6
or ten days. The revised bottom path, still the critical path, would be 5 + 6
or 11 days.
As a control device, a PERT program focuses attention on the most im
portant areas for management action — the steps in the critical path. Small
delays on slack or noncritical paths usually are not significant enough to
affect the final completion date. Delays on the critical path, however, require
immediate attention and effective followup if the target completion date is
to be met.
Chapter 6 presents a case example indicating the method for defining
the critical path. Careful study of this article will result in excellent ground
ing in the basic PERT technique. Also included in Chapter 6 is an article
with a broader discussion of the PERT technique and its applications.
Sometimes problems exist which defy solution by standard techniques.
A subset of scheduling problems called "sequencing" or "line balancing"
problems often fall into this category. A general method called heuristic
programming or, more simply, heuristics, has been found very useful in
attacking problems of this type. The strict definition of heuristics is, literally,
that it "serves to find out and encourages further investigation." When
applied to production problems, heuristic techniques lead to solutions by
trying "common sense" rules and procedures rather than rigorous optimality
criteria. Heuristic results are usually not optimal; but, since optimal solu
tions are very difficult or impossible to find in some problems, heuristic
solutions are very useful.
Chapter 7 applies heuristics to a typical production problem. We might
suggest that whereas the conventional techniques described early in Chapter
1 are quite popular, there is strong evidence that certain formal techniques
are often more likely to lead to practical, rigorous answers to planning and
control problems. These formal methods include linear programming, linear
control rules, PERT, and heuristics, all of which are thoroughly examined
in Chapters 2 through 7.
Chapter 2
PRODUCTION PLANNING
WITH LINEAR PROGRAMMING
Do You Want Production or Profit?*
Nyles V. Reinfeld
Manufacturing management is often in a position where it can choose be
tween two roads of action — increase production, or increase profits. The
first does not necessarily provide the latter. The difference between these
viewpoints was clearly pointed out in a Linear Programming study recently
made in the Tube Mill of a large midwest manufacturer.
The study showed that scheduling for maximum profit would increase
company profits by close to $350,000 over last year. 1 On the other side of
the ledger, scheduling for maximum output would increase the quantity by
22% but profits would decrease 23% below the maximum profit, or about
$300,000 a year! This increase in profit was the direct result of a selection
of products, using Linear Programming, and is based on the same produc
tion costs in both cases. The analysis considered such factors as plant and
machine capacity, sales forecasts by various warehouses, shipping rates by
items to the various distribution centers, and present and proposed com
pany policies.
* From Tooling and Production Magazine (August 1954), 16. Reprinted by per
mission of Huebner Publications, Inc.
1 All figures have been changed to retain the confidential nature of the findings.
They are, however, representative of the true picture.
13
14 PLANNING AND CONTROLLING PRODUCTION LEVELS
THE TUBE MILL STUDY
The problem in the Tube Mill consists of an overload on present produc
tion capacity. This overload creates a conflict between Standard and Capped
sizes of tubing. The result has been a tendency toward purchasing some
sizes of Standard and Capped tubing for resale through the company's vari
ous warehouses. Purchased items can be bought at a slight discount; but,
after handling costs, they are resold at just about breakeven. In other words,
there is no profit in purchasing tubing to fill orders. It is merely done as a
customer convenience. The company's customers are serviced by nine
warehouses scattered across the country. Due to the increase in shipping
costs, the farther the warehouse from the factory, the greater is the tend
ency to purchase tubing from outside sources close to the warehouse.
The method of supplying tubing to the customer as originally proposed
was to manufacture everything for the home warehouse. All Capped tubing
for the remaining warehouses was to be bought and resold. Any remaining
production capacity would then be used to fill orders for Standard tubing
for nearby warehouses. All other tubing would be bought and resold. The
pattern, here, is clear. The company was following the logical pattern which
would reduce shipping costs and therefore increase profits.
At first view, this seems to be a good policy. Actually, it was not until
Linear Programming was applied that it could be shown that a far more
profitable policy was possible.
To set the problem up for Linear Programming, it was necessary to
gather a large amount of data about every machine in the plant and about
every product to be produced on the equipment. In some cases, the data
was not directly available and had to be developed.
The plant handled several thousand products which represented many
variations of approximately a hundred standard sizes of tubing. Each piece
of tubing passed through a series of operations such as the piercer, pointers,
saws, annealing furnaces, draw benches and straighteners. In many cases,
it passed through some operations several times. On the draw benches, for
example, some pieces of tubing required as high as 13 separate draws in
bringing them down to size.
BASIC PRINCIPLES
Clearly, there are machines and factors that do not play a material role
in determining which products should be produced and which should be
bought. One of the problems, therefore, in formulating the attack, is to
determine which data is essential.
The philosophy that underlies the application of Linear Programming to
the Tube Mill problem is based on the concept of a bottleneck. A bottle
DO YOU WANT PRODUCTION OR PROFIT? 15
neck is a condition which limits the total plant output. The bottleneck may
be a certain operation or a group of operations or machines. In the case
of the Tube Mill, draw benches limit the tube capacity. All other functions
of that department, such as sawing and piercing, are limited in their output
by the draw benches. Most of these "beforeandafter" operations are
working six or eight hours a day, while the benches are busy continuously,
around the clock. Everything we can get through the benches, we can
easily get through the rest of the department. Therefore, the bottleneck de
termines the ultimate profit that we can make by operating the Mill.
We can increase the dollar output of the bottleneck in two ways : We can
buy new equipment and enlarge the capacity at the point of the bottleneck,
or we can increase the earning power of the present equipment.
We can relate profit to time in the bottleneck. Last year approximately
1 ,400,000 minutes of draw bench productive capacity was used on standard
tubing. If the profit from standard tubing was about $1,000,000 last year,
the company made $.714 for each minute the draw benches were operating.
If we could increase this to $ 1 .00 per minute, the company's profit for the
year would be $1,400,000.
You will soon see how it is possible to increase the profitperminute by
means of Linear Programming.
What is true of profit is also true of production. If we wish to increase
the quantity through the bottleneck, we must then work in terms of quantity
perminute. In either case, it is possible to achieve a considerable increase,
without any change in equipment. (This statement is not intended to mini
mize the importance of equipment modernization, methods improvement,
etc. — when necessary.)
To increase the profitperminute involves the concept of opportunity
profit. Opportunity profit is the increase in profit that can be made by
running one part instead of another, and is determined by the profitperpiece
divided by the piecesperminute. It is seen from this fact, that the piece
with the highest unit profit will not necessarily give the highest profitper
minute.
Let us exaggerate this concept for purposes of illustration. Suppose that
one piece of tubing makes a profit of $6.00, with a running time of 10
minutes. The profitperminute would be $.60. Another piece of tubing may
make a profit of $5.00, with a running time of 8 minutes. In this case the
profitperminute would be $.625.
In other words, if we could sell all we could make of either product, we
would make the greatest profit by making the latter item. That such cases
actually exist in Industry is shown by the solution we arrived at in the
case of our Tube Mill example (see Table I). It is more profitable to make
some items for the West Coast and pay the shipping cost than it is to
16 PLANNING AND CONTROLLING PRODUCTION LEVELS
make others for the home warehouse. The same thing is true for home
warehouse items only — namely, some prove to be more profitable than
others, without direct regard to the unit price tag attached to each.
SOLVING THE TUBE MILL PROBLEM
Working with the basic principles just discussed, the first step, after
isolating the bottleneck equipment (draw benches) was to gather the rela
tive data and organize it into a coherent picture.
Gathering the data essentially amounted to getting a list of the benches,
utilization figures, available time on each, a list of all the tubing produced
in the Mill, a standard operational breakdown for each size, and a list of all
the equipment alternates and times for making each single draw. Since time
was given in terms of draws, which represented any number of pieces, it
was necessary to determine the number of pieces that were involved in each
draw. In this way we were able to get the time per piece per draw.
A list of the warehouses, and sales forecasts by warehouses by products,
were obtained along with shipping rates and the profitperpiece at the plant
before shipping costs were deducted. Whereas profit is normally based upon
total cost variance including shipping charges, what we need is the profit
at each specific warehouse.
Shipping rates were given in terms of weight minimums and destination.
The rates for a specific destination were then deducted from the factory
profit on each product to give the profit at each warehouse.
The next step was to set up the matrix for solving the problem. (A matrix
is a mathematical arrangement of the coefficients of algebraic equations.)
Figure 1 shows a part of the simplex matrix as it appears for one size of
tubing and the three warehouses requisitioning that particular product.
The matrix is designed to convert normal profit into opportunity profit,
and to compare for profitability of producing various products. We don't
know what the opportunity profit is by direct inspection, because we don't
know what combinations of machines will be used to make the products.
The only approach is to solve all possible combinations of time, profit, and
product that are relevant to obtaining the best answer.
The matrix in Figure 1 would normally require solution by electronic
computers. It is possible, however, to perform a transformation on the data
which will convert the matrix to a method suitable for hand calculation.
The transformation is possible because of the type of data involved. The
new (distribution) matrix, Figure 2, is much simpler to work with. In some
instances, the distribution matrix can be set up directly without actual con
struction of the simplex matrix.
In constructing the distribution matrix, the vertical columns are arranged
DO YOU WANT PRODUCTION OR PROFIT? 17
in descending order by relative profit. In scheduling the utilization of avail
able capacity to meet forecast requirements, assignments are made on the
matrix, working from left to right — giving priority to those product orders
having the greatest relative profits. The complete conversion of the simplex
matrix required two weeks, at which time the answer was obtained directly
by setting up the distribution matrix, without further solution.
The rules for performing these operations are quite simple and can be
readily taught to clerical workers.
The peculiarity of the data which made this type of matrix solution
possible is the fact that there is a bottleneck within the bottleneck machine
group. This bottleneck exists on the last four machine groups. When these
machines were loaded, there was still open capacity on the first machines.
This meant that we could solve the problem for the smaller bottleneck and
then finish using our capacity on the first machines with products that can
be made on them alone.
Caution must be exercised in working with bottlenecks of this sort since
there is a tendency for the bottleneck to shift as the mix changes. In this
case, were the mix to change over a period of time to where the first
machines became the bottleneck, then the present solution would no longer
be valid. However, since this solution is for a year in advance, it would
normally be safe to assume that a regular pattern of sales would not affect
it. In the case of other plants that have been analysed, bottlenecks of this
type have shifted from week to week.
DIFFERENT SOLUTIONS
In arriving at the solutions, we approached the problem from the stand
point of the forecasted sales and also from the standpoint of both increased
capacity and expanded sales forecasts. We will not go into how these varia
tions were handled other than to emphasize the generality of the final
answers.
In arriving at the solution of what products to make and which to buy,
we ended up with a priority of products. This priority lists the most profit
able products first and the least profitable last. A part of this table is given
in Table I. It will be noted that warehouse No. 3 items are included right
down the line with home warehouse items, despite the difference in shipping
costs. It will also be noted that the profitsperpiece are not directly aligned
with the profitability as indicated by the priority.
Because of the general approach that was used, the answers obtained
cover a wide range of variation in sales and capacity. The solution is true
for the present capacity as well as the expanded, and has the additional
virtue of considering the course of action to pursue as sales increase. Since
the answer is given in terms of priority of one product over another,
18
PLANNING AND CONTROLLING PRODUCTION LEVELS
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DO YOU WANT PRODUCTION OR PROFIT?
19
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20 PLANNING AND CONTROLLING PRODUCTION LEVELS
quantity and capacity determine only how many of the products on the
bottom of the list are to be purchased. In other words, on the basis of the
load ahead of the equipment, the overload is purchased from outside sources
in order of lowest priority.
TABLE I. HIGHESTPROFIT PRIORITIES
Location Profit/Pc.
Product* (Whse) Priority /Whse
11/4B3
2
10
4.50
11/4B
1
11
2.03
11 /4A3
1
12
4.48
11/2A
2
13
2.23
11/2B
2
14
2.20
11/4B
2
15
1.85
11/4A
1
4 16
1.89
1A3
1
17
3.99
11/4B3
3
18
3.94
* A partial list of all the products, arranged in order of the priority applicable when objective is to
use present capacity for making highest profit.
TABLE II. HIGHEST PRODUCTION PRIORITIES
Product * Priority t
11 /4B 4
11 /4A 5
3/4B 6
1C 7
3/8A 8
1/2B 9
3/4A 10
* A partial list of all the products, arranged in order of the priority applicable when objective is to
increase output.
f No distinction made as to location, since profitability is not considered.
Price changes affecting profit on one single item will require special
analysis of that one product, which will probably be shifted in the priority
table. Cost changes affecting all products equally, such as increases in pay
scale, generally will not affect the status of the priority table.
Several different solutions were made for comparative purposes. These
solutions were given in terms of a list of products, such as the ones shown
in Tables I and II. Products were listed by priority and showed what should
be produced to accomplish a certain objective. In all cases for comparative
purposes, the same capacity and forecast figures were used.
Similar tables were made up showing comparisons of the highestpossible
profits with last year's production (Table III) and with the proposed
DO YOU WANT PRODUCTION OR PROFIT? 21
method of purchasing which was outlined earlier and designed to reduce
shipping costs.
TABLE III.
COMPARISON
OF
RESULTS*
Table I
{HighestProfit)
Table II
(HighestProduction )
Pieces Produced
Increase in Pieces
Profit Produced
Increase in Profit
820,000
$1,300,000
300,000
1,000,000
180,000
$1,000,000
* Based upon use of same capacity.
It was found that shipping costs are not a major factor in determining
maximum profit. This is shown by the fact that use of the highestprofit
table would increase profits by $300,000 over the original proposed policy
and by $350,000 over profits realized by last year's production. The
$300,000 profit increase constitutes the cumulative opportunity profit. This
profit is not realized by cost reduction or by increased production, but
solely by the opportunity profits of all items considered simultaneously.
CONCLUSIONS
The findings and data, that we have just discussed, are ready for use as
soon as the final report is handed in to management. In spite of the rather
complex mathematics involved, the only requirement on the part of manage
ment is that they understand the use of the tables.
With these tables, it is possible to work from either the highest profit or
the production table (depending on your objective) to determine the whole
program for the coming year. Once a policy has been established, it is a
simple matter to refer to the table when making a decision between pro
ducing or purchasing.
All the profit figures are based on present capacity and forecasts. Natu
rally, as sales and capacity are expanded, the same possibilities exist on a
larger scale.
As a followup to the profits study, Linear Programming is now being
used in the Tube Mill to build weekly schedules for maximum utilization of
equipment. This type of program ties together the allocation procedure and
the scheduling phase, which are normally treated separately.
The profits study is just one example of the many ways in which Linear
Programming may be used to increase company profits or reduce costs.
This type of approach can be applied to any industry which has the prob
lem of making outside purchases to supplement their line, or the com
22 PLANNING AND CONTROLLING PRODUCTION LEVELS
parable problem of subcontracting many of the components for an end
product.
Linear Programming is being used successfully for a wide range of man
agement problems other than direct profit studies. Examples include salary
evaluation, production scheduling, inventory control, and market research.
Applying Linear Programming to the
Plywood Industry*
Ernest Koenigsberg
The applications of linear programming to the plywood industry, in par
ticular the softwood industry, have recently been studied. This paper dis
cusses the preliminary examination, augmented by actual data wherever
possible. The general value of linear programming in these applications
having been confirmed by this early study, more extensive work is now
being carried out for specific companies.
The studies made have shown that, when costs are expressed in a simple
way in terms of some volumetric measure, the very significant contributions
of grade and other factors (sanding, glue, and so forth) which depend on
surface measure, are hidden. Costs should rather be considered, as in the
Mayhew system, in terms of the product specifications and labor content of
a plywood panel. The method of extending the cost studies that was sought
should therefore provide better rules and guides to aid in management
decisions, such as log purchases, veneer production, and product mix. Two
of these problems (product mix and log purchases) are treated here, inde
pendently of one another, to show how the use of mathematical models can
improve the profitability of a plywood manufacturing enterprise. The
models demonstrated are somewhat simplified but they are, nevertheless,
realistic and lead to meaningful and valuable results. Simpler problems were
treated because of the limitations imposed by a small computer, the absence
of necessary data for a fullscale calculation, or both. Larger problems are
now being investigated for specific companies.
* From Forest Products Journal, Vol. 10, No. 9 (1960), 48186. Reprinted by per
mission of Forest Products Journal.
APPLYING LINEAR PROGRAMMING TO THE PLYWOOD INDUSTRY 23
PROBLEM IPRODUCT MIX
Softwood Plywood Production: A mill purchases (or obtains from gov
ernment or owned land) logs that can be peeled in several thicknesses. Any
log yields veneer of 6 or more grades, the percentage yield among grades
varying from log to log even when logs of a single grade are peeled to the
same thickness. Veneer costs then are subject to a somewhat arbitrary distri
bution among the various grades (which are the "joint products"). Even
the best system of allocating veneer costs, that of assigning costs according
to the "realization" on the finished products containing the grade, is subject
to merited criticism. Another company using the same logs and peeling in
the same way could conceivably have different veneer costs if the product
mix were different. Further, the same logs could be peeled differently to
obtain still a third set of veneer costs by grade. Since the assignment of
veneer costs depends on the final product mix, these costs cannot be used to
determine an "optimal" product mix.
Consider the "X Plywood Company." In 1956, this company produced
44 products in the quantities listed in Table 1 . Each final product required
three or more sheets of veneer of the following grades and thickness:
Thickness Grades used\
1/10" A, B, C, D, Cx, Dx
1/8" A, B, C, D, Cx, Dx
3/16" C,D,Cx, Dx
f The subscript x indicates crossband.
Assume that the available logs give the mixture of grades and sizes re
quired for the listed quantities of final product. The available veneer supply
is fixed; it is listed in Table II. Since we cannot justify costing veneer by
grade and thickness, we cannot calculate a profit per unit of product. In
stead we shall use "return" as a measure of profitability. "Return" is de
fined as the selling price minus standard trade discounts and those cost
contributions (other than veneer) which are productdependent (labor,
glue, and so forth). The problem can then be expressed in the following
way:
"Which product and how much of each should be produced from a
known distribution of veneers of various grades and thicknesses so that
the return is maximized?"
It must be tacitly assumed that we can market any products manufactured
without further discounts. This latter assumption will be modified later.
Now, an A grade sheet of a given thickness can be used for any product
that requires the A grade sheet. The amount of A grade sheets is limited;
if we use the sheets for AD panels, we restrict the possible production of
AA, AB and AC panels. Similarly the use of Dx sheets in AD panels
24
PLANNING AND CONTROLLING PRODUCTION LEVELS
restricts the possible production of other products requiring the Dx sheets
of the same thickness. Hence all products are intimately connected. These
restrictions can be expressed mathematically in a simple form, and the
resulting mathematical equations can be manipulated by the method of
linear programming.
TABLE I. PRODUCTION OF PLYWOOD PANELS IN S.M.
Grade and
Size
X Company Average Company
AA
Interior
BB
Interior
AD
Interior
CD
Interior
AA
Exterior
AB
Exterior
AC
Exterior
BB
Exterior
CC
Exterior
1/4"
3/8"
1/2"
5/8"
13/4"
1/4"
3/8"
1/2"
5/8"
3/4"
1/4"
3/8"
1/2"
5/8"
3/4"
5/16"
3/8"
1/2"
5/8"
3/4"
1/4"
3/8"
1/2"
5/8"
13/4"
1/4"
3/8"
1/2"
5/8"
13/4"
1/4"
3/8"
1/2"
5/8"
3/4"
1/4"
3/8"
1/2"
5/8"
3/4"
5/16'
3/8"
1/2"
5/8"
13/4"
49,300
288,200
65,500
117,000
139,400
208,500
49,700
58,900
696,800
1,052,000
230,000
366,400
83,100
168,100
476,500
438,600
122,800
150,900
1,175,200
1,766,800
6,017,700
10,502,800
1,085,900
2,468,000
1,301,200
3,159,400
784,100
1,432,200
2,798,800
6,579,100
324,500
2,962,400
1,031,300
5,992,900
1,271,800
5,393,700
2,466,000
6,635,900
78,300
596,300
133,200
149,700
60,600
• 116,800
48,500
83,900
18,000
22,000
87,100
143,900
223,200
298,100
147,400
130,300
164,400
186,500
17,300
40,800
147,200
172,800
2,945,500
3,707,000
3,280,400
4,761,500
961,200
1,396,100
264,300
411,500
626,400
1,253,400
115,500
69,100
—
. 13,300
30,700
12,500
472,400
2,359,600
920,300
2,546,100
38,100
158,200
136,400
338,800
65,900
342,400
142,000
262,400
29,800
66,200
APPLYING LINEAR PROGRAMMING TO THE PLYWOOD INDUSTRY
25
TABLE II. VENEER
AVAILABLE IN S.M.
Grade and
X 1
n Decimal
A verage
In Decimal
Thickness
Company
Fraction
Company
Fraction
fA
9,781,000
.0786
15,750,000
.0550
B
683,000
.0055
803,000
.0028
1/10" 1
C
6,021,000
.0484
15,086,000
.0527
D
10,804,000
.0868
28,059,000
.0981
Cx
5,897,000
.0474
8,425,000
.0294
LDx
12,998,000
.1045
32,520,000
.1137
f A
15,776,000
.1268
28,119,000
.0983
B
5,179,000
.0416
12,938,000
.0452
1/8"
C
9,464,000
.0761
22,655,000
.0792
D
12,011,000
.0965
33,710,000
.1178
Cx
1,962,000
.0158
5,735,000
.0200
.Dx
7,877,000
.0633
27,549,000
.0963
f C
2,581,000
.0207
7,016,000
.0245
3/16"
D
5,656,000
.0455
11,636,000
.0407
Cx
7,108,000
.0571
13,387,000
.0468
iDx
10,632,000
124,430,000
.0854
22,742,000
286,130,000
.0795
TOTAL
1.0000
1.0000
Let Xi be the fraction of product i produced (for example i = 17 indi
cates AA % inch Interior, i = 18 indicates AA % inch Interior . . . i = 56
indicates CC % inch Exterior), and bj be the fraction of veneer / available
(/ = 1 indicates A y 10 inch, / = 2 indicates B i/ 10 inch, . . . / = 16 indi
cates D x % 6 inch). Now, the total production requiring /' grade veneer can
not exceed the available quantity of that veneer, or
(1) SayXt^bj
i
where a y is the number of sheets of / grade veneer required for product i.
For example, SL 17tl = 2, means two A y 10 inch sheets are required for each
panel of AA % inch; a 18 i = 2, two A i/ 10 sheets are required for each panel
of AA % inch, A 56 i = 0, no A i/ 10 inch sheets are required for CC % inch.
There are 16 such requirement equations connecting the 40 to 45 product
variables. These equations plus one more
(2) Z = 2R i X i = Maximum,
define the problem. Here Ri is the return on the I th product. In effect
equation (2) states that we want to maximize the return on the available
raw materials. The return on all products is listed in Table III.
Solution of linear programming problems of this order are timeconsum
ing if done by hand. Solutions have been obtained on the IBM 650 com
puter (using a modified H. Smith program) in about 30 minutes of machine
time. The solutions show that considerable savings are possible. For "X
Plywood Company" we find the return (per 1,000 S.M. of veneer) is
26
PLANNING AND CONTROLLING PRODUCTION LEVELS
$22.47; this is to be compared with $21.16 obtained from their actual pro
duction. The difference, $1.31, when applied to the annual production of
about 120,000,000 Surface Measures of veneer could result in increasing
the return by about $150,000 per year. The bulk of the increase in return
TABLE III.
IDEAL SOLUTION FOR X PLYWOOD COMPANY
Percent
of Total
"Shadow
Raw
Shadow
Product
Production
Price"
"Return"
Materials
Price
'1/4"
17
$22.42
$ 72.61
A
$35.72
Interior
3/8"
18
—
13.59
101.21
B
22.25
AA
1/2"
19
—
3.55
120.63
1/10"
C
19.81
5/8"
20
—
17.24
132.03
D
—
J/4"
21
.1078
157.93
Cx
16.32
.Dx
23.60
'1/4"
22
—
16.77
64.79
_
38.49
24.43
9.93
4.43
35.42
3/8"
23
—
14.83
85.24
A
B
C
D
Cx
AB <
1/2"
24
—
6.67
102.78
5/8"
.3/4"
25
26
5.09
6.47
129.45
136.73
1/8" «
'1/4"
27
—
7.02
52.30
Dx
33.49
3/8"
28
—
8.16
72.58
re
17.60
AD «
1/2"
29
.0810
—
90.74
~ ,^„\ D
5.32
5/8"
30
.0739
—
115.21
3 / 16 1 Cx
31.10
[3/4"
31
—
3.93
119.94
iDx
37.82
f5/16"
32
.1099
43.41
3/8"
33
.1049
—
48.69
CD 
1/2"
34
.0727
—
67.01
5/8"
35
—
9.69
76.08
3/4"
36
.0628
—
94.43
ri/4"
37
.1569
87.75
Exterior
3/8"
38
.0527
—
108.07
AA s
1/2"
39
—
4.85
124.58
13/4"
40
.0049
—
156.77
r i/4"
41
5.29
68.99
AB <
3/8"
42
—
1.55
92.42
1/2"
43
.0107
—
114.70
1 3/4"
44
—
7.45
141.30
f 1/4"
45
12.62
59.23
3/8"
46
.0100
—
79.51
AC <
1/2"
47
—
4.85
96.02
5/8"
48
—
26.47
110.38
[3/4"
49
—
1.34
126.87
r 1/4"
50
.0110
60.81
BB
5/8"
51
—
17.59
118.37
13/4"
52
.0777
—
127.32
r i/4"
53
6.84
49.10
CC 
1 3/8"
54
.0631
—
55.27
1 1/2"
55
—
12.91
79.16
1 5/8"
56
—
10.57
90.04
APPLYING LINEAR PROGRAMMING TO THE PLYWOOD INDUSTRY 27
is profit, because the fixed costs will be practically unaffected by changes in
the product mix. The form of the results of the linear programming compu
tation is shown in Table III. The "ideal" solution calls for the production of
only the 15 products listed here:
Interior grades: A A %, AD i/ 2 , AD %, CD % 6 , CD %, CD i/ 2 , CD •%.
Exterior grades: AA %, A A %, A A %, AB i/ 2 , AC 3/ 8 , BB %, BB %,
CC 3/ 8 .
The production of any other items must lead to a reduction in return.
These reductions can be discussed in terms of the third column of Table III
marked "Shadow Price." The shadow price is the loss in return per 1,000
S.M. of plywood panels produced (since this can only be produced by re
ducing the production of at least one of the ideal products). In this case,
the production of 1,000 S.M. panels of interior AA 14 inch would produce
a loss in return of $22.42 while the production of 1,000 S.M. of exterior
AB % would only produce a loss in return of $1.55. The value of the
shadow prices is that they allow one to determine how the return is affected
by deviation from the "ideal" product mix.
The "solution" of the problem consists of more than the quantities to be
produced and the shadow prices. We also obtain a matrix or table of num
bers that shows the best way (that is, the cheapest) to introduce a product
not included in the basic solution. This is demonstrated by example later.
Further, by using the shadow prices we can determine if a new product is
profitable or if it is profitable to make an established product in another
manner.
The solution here is not completely practical in that market restrictions
have been ignored. For example, about 1 1 per cent of production is interior
AA %. The market for this product is relatively small. Similarly, there is
no production of exterior BB %, which is a popular product. These re
strictions can be incorporated in the original set of equations, in which case
we would have found a different solution (and, of course, a lower value of
the return). We can, however, within a fairly wide range use the shadow
prices and the final "matrix" to determine the effect of modifying the pro
duction. For example, our solution results in an excess of 2.6 S.M. of y 10
inch D veneer per 1,000 S.M. of initial product. This could be used as
14 mc h Dx and give a slight increase in our return (increased by $23.60
X 0.0026 = $0.06, making the return $22.53).
The value of $23.60 is the shadow price associated with y 10 inch Dx
veneer. The shadow price of the raw material is the added return that can
be obtained if there were an additional 1,000 S.M. of the raw material
28 PLANNING AND CONTROLLING PRODUCTION LEVELS
available. These shadow prices are indications of how peeling can be modi
fied in order to produce greater returns, or how one might improve the
return by changing the quantity peeled in each thickness. A study of Table
III, for example, shows that Dx has a greater shadow price than D for all
thicknesses. In an actual problem a result of this form would indicate that
downgrading D to Dx may be profitable. A large difference in shadow price
between B and C grades would indicate that more labor on patching grade
C to B might be profitable. The differences in shadow prices also give indi
cations of changing the peeling distribution.
Table IV shows the solution obtained for the "Average Company." The
"ideal" return for the Average Company is $21.22 as compared with the
actual return of $20.08 per 1,000 S.M. of veneer. We obtained $.2504 units
of product for each unit of veneer. This is a difference of $1.14 per 1,000
S.M. of veneer produced. The ideal production for this company differs
from the previous example for two reasons: (1) the return (and cost)
structure is different (Tables III and IV) and (2) the raw product mix is
different (Table II). X Company uses logs that yield more A and B grade
and also peel more % 6 inch, hence the greater return.
We can examine the changes implied by altering the production. Let us
assume that we must produce interior AB % inch (product 26). To do this
we must change our production of other products. The final matrix for
product 26 gives the values shown in Table V, Column 1. The values show
that, for each unit of product 26 produced, the output of product 40 is
increased by 0.5 units, the output of product 52 is reduced by 0.5 units,
and the output of product 21 is reduced by 1.0 unit. (A plus sign in the
matrix column indicates a reduction and a minus sign indicates an increase.)
For each 1,000 S.M. so changed, the shadow price tells us that we will lose
$6.47 in return. If we do this for 4 per cent of production our return will be
reduced by (0.04) X (0.2504) X 6.47. (0.2504 is the number of 1,000
S.M. panels produced per 1,000 S.M. of veneer.) Thus the return is re
duced by about six cents per 1 ,000 S.M. of veneer. Column 3 of Table V
shows the final production if this change is made.
We can make further modifications based on the information generated
by our final matrix. Suppose that in addition to interior AB % inch we
must produce exterior BB % inch (product 51). The 5th column of
Table V shows the final matrix column for product 51. Each unit of
product 51 produced results in an increase of 1 unit of product 38 and a
decrease of 1 unit each for products 46 and 40. Now, since we only produce
APPLYING LINEAR PROGRAMMING TO THE PLYWOOD INDUSTRY
29
1 percent of product 46, we cannot produce more than 1 percent of ex
terior BB % inch (or else we would have a "negative" production). If we
TABLE IV.
SOLUTION FOR AVERAGE PLYWOOD COMPANY
Percent
of Total
"Shadow
Raw
Shadow
Product
Production
Price"
"Return"
Mate
nals
Price
r \/4"
17
$10.57
$ 81.16
A
$39.92
Interior
3/8"
18
—
8.31
101.22
B
26.34
AA
1/2"
19
—
.65
120.34
1/1 6"<
C
21.65
5/8"
20
—
2.21
132.99
D
11.45
J/ 4"
21
3.09
151.67
Cx
10.31
,Dx
11.90
'1/4"
22
—
10.70
67.45
3/8"
23
—
8.11
84.64
A
42.87
AB 
1/2"
24
.0682
—
104.20
B
26.09
5/8"
25
.0757
118.42
1/8" 
C
20.27
[3/4"
26
—
2.97
135.00
D
Cx
14.61
19.48
fl/4"
27
—
9.71
53.56
Dx
14.01
3/8"
28
—
7.85
73.42
rc
19.30
AD <
1/2"
29
—
.18
92.55
D
21.43
5/8"
30
.1023
—
106.95
3/16"<
Cx
28.15
3/4"
31
—
2.63
123.87
LDx
23.79
'5/16"
32
.0980
45.00
3/8"
33
.0105
—
48.90
CD <
1/2"
5/8"
3/4"
34
35
36
.1315
.0273
.1739
—
68.35
77.53
97.08
fl/4"
37
.1203
90.15
Exterior
3/8"
38
.0105
—
113.89
AA "
1/2"
39
—
.84
127.18
13/4"
40
.0808
—
161.97
'1/4"
41
_
.49
76.08
AB <
3/8"
42
—
1.95
95.16
1/2"
43
.0011
—
111.23
.3/4"
44
—
—
145.89
'1/4"
45
12.08
59.80
3/8"
46
—
8.92
82.37
AC <
1/2"
5/8"
47
48
—
5.63
6.32
99.78
115.71
.3/4"
49
—
5.39
133.98
fl/4"
50
.0061
—
62.99
BB
5/8"
51
.0101
—
111.06
13/4"
52
.0163
—
128.40
fl/4"
3/8"
1/2"
1 5/8"
53
3.14
50.47
CC <
54
55
.0674
4.92
60.02
80.65
56
—
8.01
91.76
30
PLANNING AND CONTROLLING PRODUCTION LEVELS
TABLE V
Matrix
Ideal
Matrix
Column
Production
Column
Product
{Percent
Modified
Product
Modified
Product
26
of Total)
Production
51
Production
Interior
CD
3/8"
33
.1049
.1049
.1039
Exterior
BB
1/4"
50
.0110
.0110
.0110
Interior
CD
5/16"
32
.1099
.1099
.1099
Exterior
AA
3/8"
38
.0527
.0527
1.000
.0627
1/4"
37
.1569
.1569
.1569
Interior
CD
1/2"
34
.0727
.0727
.0727
AD
1/2"
29
.0810
.0810
.0810
Exterior
AC
3/8"
46
.0100
.0100
1.000
CC
3/8"
54
.0631
.0631
.0631
D 1/10*
4
Interior
AD
5/8"
30
.0739
.0739
.0739
Exterior
AA
3/4"
40
.5000
.0049
.0249
1.000
.0149
AB
1/2"
43
.0107
.0107
.0107
BB
3/4"
52
.5000
.0777
.0577
.0577
Interior
CD
3/4"
36
.0628
.0628
.0628
AA
3/4"
21
1.000
.1078
.0678
.0678
AB
3/4"
26
—
.0400
.0400
Exterior
BB
5/8"
51
.0100
Shadow price
$6.47
$17.59
Total return
$22.47
$22.41
$22.37
* Unused raw material.
produce 1 percent of exterior BB % inch we end up with the production
listed in the last column of Table V. The return is reduced by (0.01) X
(.2504) X 17.59 = 0.04.
The process can be carried on to include as many essential products as
necessary. This substitution process is extremely useful for determining
the return for various combinations of production. The changes made in
production quantities are those which would introduce the products at
minimum loss in return. The alternative to this method is the reevaluation
with added market condition equations. The program can be set up so that
the effect of price changes can be treated starting from an existing solution.
The latter requires much less computer time for the new calculations. One
can define condition equations for the size of price change necessary to
require a new solution. The method is extremely flexible to almost all
possible changes.
APPLYING LINEAR PROGRAMMING TO THE PLYWOOD INDUSTRY 31
The example shown here illustrates that improved production perform
ance is possible. The tables also show the data for an "Average Company,"
the data for which has been generated from a cost study of some dozen
companies for 1957. The results for the average company show savings
of the same order of magnitude as the example company. We have carried
out other calculations with the same sort of results. In our calculations,
we were restricted somewhat by the available data and more by the re
strictions in the computer program. Because of the latter limitation we only
considered 41 final products. A more extensive program for the IBM
650 is available that will allow about 55 products and 20 raw material and
market restriction equations. For largerscale problems, the IBM 700 series,
the UNIVAC, DATATRON, and other computers are available. If such
machines are used there would be no need to omit such factors as sales
potential, interchangeability of raw materials, alternate methods of making
the same final products, and so forth.
Hardwood Plywood Production: No thorough analysis of hardwood ply
wood production has been made thus far. The differences between hard
wood and softwood production are those of complexity and variety rather
than differences in principle. In hardwood, one deals with a number of
varieties of wood in a range of grades rather than a single variety. Logs
are also purchased in a greater multiplicity of log grades. The complexity
introduced by the numerous varieties and grades, and the fact that the
products and veneers have much higher values, makes the selectivity, that
is, the decision on product mix, a more critical problem.
The basis of any analysis of production mix, by purchases, veneer peel
ings, and so forth, is a good cost system. Without knowledge of the costs
of various alternatives one can make only rough guesses as to the relative
value of a given production, peeling or purchasing policy. Such cost systems
are not firmly established in the hardwood industry and examples are there
fore hard to come by. We shall, however, discuss the application of linear
programming methods to a number of possible problems in the manufacture
of hardwood panels.
First, let us consider the softwood producer who also manufactures hard
wood panels with softwood centers. The hardwood panels "compete" for
softwood centers and cores and thus interact with all the softwood products.
If a Cx core is used for % inch AC Exterior panel it is not available for a
% inch Birch panel; a B grade center used for a hardwood panel restricts
the production AB and BB products that require the same thickness sheet
of veneer.
Assuming that the variable costs of hardwood panels are known, we can
treat the hardwood panels in the same way as other products. We shall
consider two cases.
32 PLANNING AND CONTROLLING PRODUCTION LEVELS
1 . Hardwood veneer purchased outside and no limit on available supply
(that is, a free market with adequate sources of veneers).
In this case the problem can be set up in the same way as problem 1 . The
various hardwood products are introduced only through the softwood com
ponents of the layups. The return on a hardwood product is defined as
Ri = (Selling price) — (discounts) — (variable costs of manufacture)
— (cost of the hardwood veneers).
2. Hardwood veneers manufactured by the company or purchased in a
restricted market (that is, the quantity of hardwood veneer in each grade
is limited).
This case differs from the example above in that the hardwood panels
"compete" for both softwood and hardwood components. If the hardwood
veneers are purchased, then the return R { is as defined above. If hardwood
veneers are manufactured, then we use the earlier definition of R { given in
Problem 1. In both cases one must introduce the hardwood components of
the layups in establishing the matrix. We will then have equations of the
form of Eq. 1 for the hardwood veneer supply as well as for the softwood
veneer supply. The hardwood products may also be included in or subject
to market restrictions of the same form as have already been presented.
Thus, the introduction of hardwood veneers and panels does not alter the
problem in any serious manner, when one is concerned with the production
of combined hardwood and softwood panels. One would expect this to be
true, since the production method is similar and the interactions are not
different in principle. In fact, the principles are even much the same for
plants which produce entirely hardwood panels, as we shall show.
Let us now consider a company that produces only hardwood panels
from a number of different species (oak, birch, maple, white pine, lauan,
and so forth). Centers and cores are obtained from the hardwood (or
perhaps by purchases of softwood panels or veneers). As in the softwood
case, there are a number of different raw materials (veneers by species,
grade and thickness) obtained by peeling logs. These are to be used to
produce plywood panels of various thicknesses and quality in such a way as
to maximize profits, subject to the limitations of marketability of products,
plant capacity and other plant restrictions. One can set up a layup and
restriction "matrix" just as in the case of softwood production. Because of
the larger variety of raw materials (veneers) and finished products one will
have a "larger" problem for a single plant. Within limits the size of the
problem is no obstacle; problems involving a number of integrated plants
in one system are already being studied. Given a good cost system, this
problem can be treated quite readily and perhaps yield even greater in
creases in return than in softwood production because of the higher costs
of peeling and grading in the hardwood industry.
37
11
32
20
28
9
33
30
17
8
34
41
8
7
44
41
APPLYING LINEAR PROGRAMMING TO THE PLYWOOD INDUSTRY 33
PROBLEM 2— LOG PURCHASES
In the previous example, we were concerned with obtaining the best
mix of final products from a group of sheets of veneer. This is equivalent
to obtaining the best product mix from a set of logs. The connection lies in
the relationship between grades produced from a log of given quality. A
study made by the U.S. Forest Service 1 indicates that such relations exist.
For Southwest Oregon the report gives the results below. 2
% A % B % C % D
Log Grade Veneer Veneer Veneer Veneer
# 1 peeler
#2 peeler
#3 peeler
Special peeler
Given these values and a panel recovery ratio (panels % inch equivalent
per board foot of lumber) we can evaluate the number of sheets of veneer
of each grade from the logs purchased. The DFPA Manual lists an average
industry recovery ratio of 2.2. Converting this to % i ncn equivalent veneer
we have a veneer recovery ratio of 6.6. We can use the figure above
in an example to show how the logs purchased tell us what veneer is
available and the average cost per 1 ,000 board feet.
Suppose we purchase 1,000, 1,000, and 2,000 M board feet of #1
peeler, #3 peeler, and special peeler, respectively. The number of M %
inches sheets obtained in A grade is
(1,000 X 0.37 + 1,000 X 0.17 ^ 2,000 X 0.08) 6.6 =
(370 + 170 + 160) 6.6 = 4,620 M sheets of % inch A.
Similarly, we have
6.6 (1,000 X 0.11 + 1,000 X 0.08 + 2,000 X 0.07) =
2,178 M sheets % inch B,
6.6 (1,000 X 0.32 + 1,000 X 0.34 + 2,000 X 0.44) =
10,614 M sheets % inch C,
6.6 (1,000 X 0.20 + 1,000 X 0.41 + 2,000 X 0.41) =
9,438 M sheets % inch D.
1 U. S. Forest Services, Report R6, "West Side PlywoodLumber Appraisal Base,"
Table 8 (October 22, 1957).
2 The results are the average for large and small logs rounded off to the nearest
percent.
34 PLANNING AND CONTROLLING PRODUCTION LEVELS
If, for the log cost, the listed values for the Example Plywood Co. in the
DFPA Manual, are used:
#1 peeler $90/M bd. feet
#2 peeler $80/M bd. feet
#3 peeler $70/M bd. feet
Special peeler $53.72/M bd. feet
we obtain for the average log cost/M board feet
1,000 X 90 + 1,000 X 70 + 2,000 X 53.72
Average cost =
4,000
$66.86/M bd. feet.
Recognizing the fact that the purchased logs define the distribution of
veneer by grades (assuming, at this stage, no substitution of one grade for
another), we can define the second problem:
"Given a distribution of final products (and hence the required num
ber of veneer sheets in each grade), what is the best combination of
logs to purchase; that is, how many #1 peelers, #2 peelers, #3 peelers,
and special peelers should be purchased?"
Therefore, this problem is just the reverse of the first problem that was
treated by linear programming techniques.
Assume that we want to satisfy the demands for y 10 inch veneer listed
in Table VI. For i/ 10 inch veneer the veneer return ratio is 8.25 (6.6/0.8;
see DFPA Manual). The log prices are those given previously and we want
to minimize the cost of logs to meet our demands. Let Pi be the quantity
of #1 peeler, P 2 the quantity of #2 peeler, P 3 the quantity of #3 peeler,
and P 4 the quantity of special peeler. We need 10,591 M sheets of y 10
inch A. Then:
8.25 (0.37Pi + 0.28P 2 + 17P 3 + 0.08P 4 1) g 10,591
(A) or 3.069Pi + 2.32P 2 + 1.40P 3 + 66P 4 ^ 10,591
Similarly, for the other grades we obtain
(B) 0.91P! + 0.74P 2 + 0.66P 3 + 0.58P 4 ^ 627
(C) 2.64Pi + 2.72P 2 + 2.8IP3 + 3.63P 4 ^ 11,905
(D) 1.65?! + 2.48P 2 + 3.38P 3 + 3.38P 4 ^ 25,531
(Note: In the absence of more complete data we have lumped the re
quirements for C and Cx and D and Dx.) We want to minimize the total
cost of logs: that is, find the values of P l9 P 2 , P3, and P 4 which satisfy
APPLYING LINEAR PROGRAMMING TO THE PLYWOOD INDUSTRY 35
equations (A), (B), (C), and (D) and minimize the cost K given by
(E) K = 90?! + 80P 2 + 70P 3 + 53.72P 4 .
This, too, is a problem in linear programming that is solved more simply
by using the socalled "dual." 3
TABLE VI. AVERAGE PLYWOOD COMPANY
1/10" Veneer in Products Using
Only 1/10" Veneer (in 1,000 S.M.)
Grade
A 10,591
B 627
C 7,938
D 12,318
Cx 3,967
Dx 13,213
Because of the small amount of B required, any solution of the problem
as stated must result in an excess quantity of B. Since A can be (and often
is) downgraded to B grade, and in most cases there is an excess of C
grade, we will solve a simpler problem; that is, consider only two grades,
A 1 = surface grade, either A or B, and C 1 == back grade, either C or
D. In this case our equations become
(A 1 ) 3.97P 1 + 3.06P 2 + 2.06P 3 + 1.24P 4 ^ 11,218
(C 1 ) 4.29Pi + 5.20P 2 + 6.19P 3 + 7.01P 4 ^ 37,436
(E 1 ) K = 90Pi + 80P 2 + 70P 3 + 53. ,72P \.
Our solutions are
Pi= 1,435 Mbd. feet
P 2 =
P 3 =
P 4 = 4,535 M bd. feet
The total cost is
K = $368,500
and the average lumber cost is
Average cost = 368 ' 500 = $61.80/M bd. feet
5 5,970
If more A 1 grade is used, the average cost will increase; the average cost
increases with increasing ratio of A 1 /(A 1 + C 1 ) and decreases with in
3 For technical reasons we cannot follow the same techniques as in the previous
example. We make use of the fact that every minimizing problem has a "dual" which
is a maximizing problem. The solutions are related in a simple manner and the total
cost is identical for the problem and its dual. See, for example, Gass, S., Linear Pro
gramming, McGrawHill, New York (1958).
36 PLANNING AND CONTROLLING PRODUCTION LEVELS
creasing ratio of CVC^ 1 + C 1 ). We can illustrate this with another ex
ample drawn from the data for the Example Plywood Company (DFPA
Manual).
The total requirements of y 10 inch veneer are listed in Table VII (these
include y 10 inch with % inch and % 6 inch veneer). In this case, the equa
tions are
(A 1 , y 10 ) 3.97?! + 3.06P 2 + 2.O6F3 + 1.24P 4 ^ 11,218,
(C 1 , y 10 ) 4.29?! + 5.20P 2 + 6.19F 3 + 7.01P 4 ^ 49,437,
(E 1 , y 10 ) remains unchanged. The solution is
Pi = 750 M bd. feet P 3 =
P 2 = P 4 = 6,570 Mbd. feet
The total cost is
and the average cost is
K = $432,000,
Average cost = or $59.00, about $2.80 less per M bd. feet.
7,320 F
We have examined the solutions for the other thicknesses. Since there
is no 3 / 16 inch surface grade (A or B), one can use only special peelers.
There will be considerable downgrading, that is 15 percent of the output
is surface grade that will be used as C grade, and some of the C grade
will be used as D grade, but the average price is not affected. For % 6 inch
the average cost is therefore $53.72/M bd. feet.
The y 8 inch veneer differs from the y 10 inch in that a large percentage
of surface grade is required — about 40 percent. This must increase the
cost of logs and hence the cost of veneer. For % inch veneer, the veneer
recovery ratio is 6.6 and, using the data from Table VII, we get
(A 1 %) 3.17Pi + 2.44P 2 + 1.65P :5 + 0.99P 4 > 35,450
(C 1 y 8 ) 3.43Pi + 4.16P 2 + 4.96P.S + 5.62P 4 > 53,450
Equation E 1 remains unchanged. The solutions are
Pi = 10,250 M bd. feet P 3 =
P 2 = P 4 = 3,200 M bd. feet
The total cost is
K= $1,119,000
The average cost is
Average cost = hi}!' — = $83.00/M bd. feet
13,450
APPLYING LINEAR PROGRAMMING TO THE PLYWOOD INDUSTRY 37
Thus the lumber costs of % inch veneer are almost 50 percent more than
those of !/, inch veneer. This is not suprising, since in % inch 40 percent
of the veneer is surface grade, while only 17 percent of the y 10 inch
veneer is surface grade. It does suggest, however, that wherever possible
alternate layups be used (assuming, of course that the veneer used for
the products by the Average Plywood Company represents exactly what
is desired for a customer and/or profit point of view) such that y 10 inch
surface grade sheets replace % inch sheets and the thicker sheets be of
the lower grades.
For the total veneer supply we have the following results: 4
1/10
inch
1/8
inch
3/16
inch
60,104
7,285
88,883
13,467
41,109
9,343
Total veneer required ....
Logs used (M bd. feet) . .
Average cost of logs
(per M bd. feet) $59.00 $83.00 $53.72
7,285 (59.00) + 13,467 (83.00) +9,343 (53.72)
Average lumber cost =
7,285 + 13,467 + 9,434
2,049,481.96
30,095
Average log cost = $68.10/bd. feet.
The application of linear programming to the log purchase problem is
more limited than in Problem 1 above because:
(1) There is a relatively small "free" market in logs; purchases are
mainly by contract for a tract;
(2) There has been far too little analysis for grade recovery from logs
of various grades; more precise data, including the variations in grade
recovery by thickness of peeling, are required.
Should the second restriction be overcome, the linear programming
technique can provide a measure of the recovery value of a timber tract,
given good survey values.
CONCLUSIONS
The value of linear programming to plywood enterprises extends be
yond providing the solutions to specific problems. As byproducts of the
solution, the technique generates information on such decisionfactors as:
( 1 ) The cost or loss in return due to a capacity or market restriction
and hence the value of new equipment or extended sales promotion (of
4 The solutions are given in roundedoff numbers obtained from sliderule calcula
tions. The values listed here are from Table IV.
38
PLANNING AND CONTROLLING PRODUCTION LEVELS
TABLE VII. AVERAGE PLYWOOD COMPANY
Total Veneer Production
By Size and Grade
Veneer
Veneer
Veneer
in
in
in
Grade
1,000
S.M.
1,000
Bd. Feet
1,000
S.M.
1,000
Bd. Feet
1,000
S.M.
1,000
Bd. Feet
Factor 8.25
Factor 6.6
Factor 4.4
A
B
C
D
Cx
Dx
10,591
627
9,347
15,000
6,784
18,303
1,283.76
76.00
1,132.97
1,818.18
822.30
2,218.54
20,706
14,761
15,252
17,535
8,815
11,814
3,137.27
2,236.51
2,310.90
2,656.81
1,335.61
1,790.00
0
0
7,278
7,667
10,873
15,290
1,654.09
1,742.50
2,471.13
3,475.00
TOTAL 60,652
7,351.75
88,883
13,467.10
41,108
9,342.72
specific products) which would cancel the restrictions.
(2) The cost of producing certain "unprofitable" products to satisfy
demands or enter a market. These costs may be too high to merit reten
tion of the customer or the market.
(3) The additional return that might be obtained by changing the
peeling distribution or by changing the log distribution.
(4) The profitability or loss (in terms of production costs) of adding
new products to the line or changing the layup for an existing product.
(5) The cost of minor adjustments to the ideal product mix and the
range in which changes can be made without changing the total return.
(6) The effects of price changes or production costs on the product
mix and on total realization.
All of these factors are of great importance to production and sales
management. The existence of this information (which is not at present
available in most companies) enables better operating and marketing de
cisions to be made. It is, in fact, possible that the availability of such
information will prove of even greater value to the plywood manufacturer
than actual solutions that linear programming can provide.
Chapter 3
MATERIALS PLANNING WITH
LINEAR PROGRAMMING
ill.
A Linear Programming Application to
Cupola Charging*
R. W . Metzger and R. Schwarzbek^
Least cost cupola charging can be accomplished with a relatively new
analysis tool called linear programming. The purposes of this presentation
are to:
1. Present some general information about linear programming.
2. Describe the cupola charge problem and the required linear pro
gramming formulation.
3. Describe how linear programming has been successfully used in a
production foundry making cast and malleable iron.
No attempt will be made to describe any of the methods of linear pro
gramming. While the problems discussed are concerned with making cast
or malleable iron, the general approach is equally valid for any blending
type problem, i.e., any problem where a variety of items are mixed,
blended, or melted to form an end product.
* From The Journal of Industrial Engineering, Vol. 12, No. 2 (1961), 8793. Re
printed by permission.
t Based upon a presentation to the Saginaw Valley Chapter of AIIE, April 19, 1960.
39
40 PLANNING AND CONTROLLING PRODUCTION LEVELS
INTRODUCTION
Linear programming can best be defined as a group of mathematical
techniques that can obtain the very best solution to problems which have
many possible solutions. While a great many industrial problems fit this
category of having many possible solutions, linear programming is not
a magic panacea which will solve all problems. However, it can be used
to solve a variety of industrial problems. For example, consider a four
plant manufacturing system which manfactures an item and then ships
it prepaid to a number of customers.
Historically, these four plants operated independently of one another.
Linear programming was applied to the distribution of the product from
these plants to the customers on the basis of the total four plant system,
and resulted in a new distribution pattern.
This solution or pattern of shipping from the plants to the customers
presents the lowest cost shipping program. If management deviates from
this plan, the total costs would increase. For example, if management
decided not to ship from the plant in Washington to the customer in Maine,
but rather to ship to him from the plant in Pennsylvania, it would mean:
One or more changes in the shipments from the factory in Pennsylvania.
Several changes in the shipments from the factory in Indiana.
Several changes in the shipments from the factory in Kansas.
One or more changes in the shipments from the factory in Washington.
The net effect of all of the required shipping changes would result in
increased total shipping costs.
This solution also illustrates the situation where the very best solution
is not necessarily the one that would be obtained by "cutandtry" or
intuitive methods. This is sometimes the case when solving a problem with
linear programming.
The methods of linear programming present a stepbystep approach
which, when followed, will arrive at the best solution. Not only is the
best solution obtained, but information is provided which permits a rather
quick analysis of the lessthanbest or alternative answers. In the pre
ceding problem, if management said that they did not wish to distribute
from Washington to the customer in Maine, the linear programming solu
tion immediately shows how much the total costs will increase and in
dicates all the prescribed changes that must be made to obtain the less
thanbest solution.
In problems like this, standard analytical procedures fail because all
facets of the problem are so highly interrelated. In these kinds of problems,
it soon exceeds human capabilities to be able to consider all facets of
A LINEAR PROGRAMMING APPLICATION TO CUPOLA CHARGING 41
the problem at the same time. Linear programming, however, is a tool of
analysis which can and does consider every facet of the problem simul
taneously.
THE CUPOLA CHARGE PROBLEM
The problem of charging or loading materials into a cupola is one which
has many possible solutions. One of the basic problems in charging
materials into a cupola, in making either ferrous or nonferrous alloys, is
in determining how much of the available materials to charge in order to
obtain the proper chemical and metallurgical properties at the lowest pos
sible cost. Linear programming can be used to obtain the lowest cost
charge which meets all the specifications of the melt.
In order to illustrate the problem and to show the results that were
obtained with linear programming, two typical cupola charging problems
will be discussed. One is typical of cupola charging in a high volume produc
tion foundry while the other is more typical of a smaller or custom foundry
operation.
The linear programming analysis requires much the same information
as presently used in cupola charging, namely:
1 . A list of charge materials, their chemical analysis and cost per ton.
2. Chemical specifications of the charge (percent range for each element).
3. Any additional restrictions on the usage of the various charge materials.
The last item of information will include limitations on the use of certain
materials either because of limited supplies or because of metallurgy
requirements.
Before we consider a specific problem it may be advisable to examine
the general linear programming approach to the cupola charging problem.
In order to use linear programming one must first describe his problem
mathematically, in a set of equations or formulas.
To illustrate this, assume we have several materials that can be put into
a cupola and melted to make grey or malleable iron. We will call them
materials Number 1, Number 2, etc. Since we wish to determine how much
of the materials to charge, we can let:
*i —  percent of charged material Number 1 actually used.
x 2 = percent of charged material Number 2 actually used.
•
x n = percent of charge material Number n actually used.
Then, what we want to do is calculate values for x lt x 2 . . . x n which will
meet all the specifications of the charge at a minimum cost. Each specifica
42 PLANNING AND CONTROLLING PRODUCTION LEVELS
tion takes the form of an equation or inequality. For example, suppose we
consider the specifications for carbon obtained from the materials in
cluded in the charge. We can say:
axx + bx 2 + cx% + • • ■ nx n ^ maximum carbon (percent)
where a, b, c, etc., are the the percentages of carbon in each material
1, 2, 3, etc. This relationship says that the grand total of carbon which is
contained in the various materials included in the charge cannot exceed
the maximum carbon specification. A similar relationship must be ex
pressed for the minimum carbon content namely:
ax x + bx 2 + cxs f • • • n *n ^ minimum carbon (percent)
These two relationships together make certain that the carbon specification
will be satisfied.
In a similar manner, we can express the limitations for all the other
elements like silicon, manganese, chrome, etc. In this manner, we can con
sider any number of chemical elements.
In developing a cupola charge, certain of the charge materials may be in
limited supply and it may be undesirable to use more than a limited
amount of other materials for various metallurgical reasons. These re
strictions must also be included in the mathematical statement of the prob
lem. For example, suppose that material Number 5 (* 5 ) is in limited supply
and we only have enough to permit charging 30 percent. Therefore, we
want to limit x 5 to a maximum value of .3 which can be expressed:
x 5 ^ .3.
In this way, we will permit no more than 30 percent of material Number
5 in the charge, if indeed it is profitable to use any of it at all. Similar
restrictions can be placed on other materials in limited supply as well as
those materials we wish to limit in the charge for metallurgical purposes.
A final equation is required in order to make certain that the solution
gives percent of the various charge materials. The relationship:
Xi + x 2 + x 3 + x 4 + . . . x n = 1
Finally the formulation requires an objective. In this case our objective is
minimum cost. Hence we include:
px 1 + q*2 + rx s + • • • = minimum .
where:
p = the total cost per ton for material Number 1 .
q = the total cost per ton for material Number 2.
•
A LINEAR PROGRAMMING APPLICATION TO CUPOLA CHARGING
43
Now the formulation or mathematical statement of the problem includes:
1 . The system of relationships describing the problem.
2. The desired objective.
In this way, then, the mathematical statement of the problem describes
or defines the problem and its limits. What linear programming does is
to seek the best solution in terms of the objective within the limits de
scribed by the system of relationships.
A PROBLEM
In order to better illustrate this formulation consider the following prob
lem.
A metallurgist has this problem:
Size of total charge
Amount of sprue and iron briquettes charged
Amount of "other material" to be charged
This 2000 lbs. of "other material" must contain:
5000 lbs.
300 lbs.
2000 lbs.
A t Least
No More Than
60 lbs. carbon
54 lbs. silicon
27 lbs. manganese
6 lbs. chrome
3.0 %
2.7 %
1.35%
.30%
70 lbs. carbon
60 lbs. silicon
33 lbs. manganese
9 lbs. chrome
3.5 %
3.0 %
1.65%
.45%
The materials to be considered are:
Manga
Cost/
Carbon
Silicon
nese
Chrome
Ton
Percent
Percent
Percent
Percent
Xi Pig Iron
$ 60
4.00
2.25
.90
x 2 Silvery Pig
129
15.00
4.50
10.00
x 3 FerroSilicon Number 1
130
45.00
Xi FerroSilicon Number 2
122
42.00
x 5 Alloy Number 1
200
18.00
60.00
x Alloy Number 2
260
30.00
9.00
20.00
Xi Alloy Number 3
238
25.00
33.00
8.00
x 8 SiliconCarbide
160
15.00
30.00
x Steel Number 1
42
.40
.90
jcio Steel Number 2
40
.10
.30
xu Steel Number 3
39
.10
.30
Maximum usages (either due to limited supply or due to metallurgy re
quirements) :
44 PLANNING AND CONTROLLING PRODUCTION LEVELS
Silicon carbide, 20 pounds per charge
Steel Number 1 , 200 pounds per charge
Steel Number 2, 200 pounds per charge
Steel Number 3, 200 pounds per charge
The problem is then to determine how much of which materials (*i
through jcn) should make up the 2000 pounds at a minimum cost.
The problem must be stated mathematically. For instance, the carbon
content of the input must be in a certain range. Each possible material
contains a certain amount of carbon. Therefore, one mathematical rela
tionship would be:
The sum of carbon in each material times the quantity of that material, must
be above the minimum requirement of carbon in the final blend.
Thus, in this problem the first relationship becomes:
4*i + 15* 8 + .4x 9 + Ax 10 + .l*ii ^ 3
(all numbers in percent)
where :
x 1 = percent (decimal) of pig iron included in the charge.
x 2 = percent (decimal) of silvery pig included in the charge.
*n = percent (decimal) of Steel Number 3 included in the charge.
In order to assure that the carbon does not exceed the maximum
specification we include:
4*! + 15* 8 + .4* 9 + .1*10 + .1*11 g 3.4
Similar inequalities (relationships which have other than an equal
sign) are established for all the chemical restrictions, i.e., silicon, man
ganese, and chrome.
The restrictions as to the use of these materials can be stated:
which means that * 9 (percent of Number 1 steel to be included in the
charge) cannot exceed 10 percent.
The entire problem is then stated mathematically as given below.
Minimize:
60*1 + 129x2 + 130x3 + 122x4 + 200x 5 + 260x 6 +*238x 7 + 160x 8 + 42x 9 + 40xi + 39x»
Subject to:
A LINEAR PROGRAMMING APPLICATION TO CUPOLA CHARGING 45
(Carbon) 4«i + 15xh + .4x y + .\x w + .lx„ ^ 3.00
,„. f . , '„, „ 4x t +15x* + .4xo+ .1*10+ .l*u^3.5
(Silicon) 2.25*! + 15x 2 + 45x 3 + 42x 4 + 18x., + 30x a + 25x 7 + 30x* ^ 2.70
,,_ , 2  2 ^x + 15x 2 + 45x 3 + 42x 4 + 18x, + 30x 6 + 25x 7 + 30x* ^ 3.0
(Manganese) .9x x + 4.5x 2 + 60x, + 9x G + 33x 7 + .9x + .3x 10 + .3x„ ^ 1.35
tr ~ , .9xx + 4.5x, + 60x, + 9x fi + 33x 7 + .9x + .3x 10 + .3x n ^ 1.65
(Chrome) 10x 2 + 20x 6 + 8x 7 ^3.0
(Total Melt) 10x 2 + 20x 6 + 8x 7  .45
Xi + X 2 + X 3 +Xi + Xs + Xe + X 7 + X* + Xo + Xw + Xn = 1.0
Maximum usages:
X, ^
.01
Xo ^
.10
Xio —
.10
Xn ^
.10
With this mathematical statement, the problem is ready for linear pro
gramming. The objective or cost equation indicates that we desire the
answer with the lowest possible cost and one which is within the limitations
described in the set of relationships. This formulation or mathematical
statement of the problem can be solved with the simplex method of linear
programming. While we will not consider the details of the simplex method
here, let's take a look at the lowest cost solution which the simplex
method provides.
The least cost solution is as follows. 1
Variable Value Description
10% Steel Number 3 in the charge
1.1% FeSi Number 2 in the charge
.711% Alloy Number 1 in the charge
3 % Silvery Pig in the charge
73.713% Pig Iron in the charge
1.472% Steel Number 2 in the charge
10% Steel Number 1 in the charge
Total Cost Per Ton $59.56.
This charge results in minimum amounts of the four elements carbon,
silicon, manganese, and chrome. As mentioned earlier linear programming
not only provides the best solution (lowest cost in the case) but it provides
information about less than best or alternative solutions. The linear pro
gramming solution for this problem also gives the following information
for those materials not in charge:
Xn
.10
Xi
.011
x 5
.00711
x 2
.03
Xi
.73713
Xio
.01472
Xo
.10
lbs. 12000 lbs
200.00
22.0
14.22
60.0
1474.26
29.44
200.00
1999.92 lbs.
1 It should be noted that the formulation of this problem can be reduced in size.
The same solution can be obtained if we omit the upper limit relationships on carbon,
silicon, manganese and chrome. In solving more than 30 similar cupola charging prob
lems for both cast and malleable iron it was found that the optimum (lowest cost)
solution always resulted in charges exactly at the lower limit specification. With this
experience it seems reasonable to formulate a cupola charge problem without the
upper limits and include these only if the solution demands them. This is a practical
approach even when high speed computing equipment is readily available.
46 PLANNING AND CONTROLLING PRODUCTION LEVELS
Material Marginal Cost
(jc 8 ) FeSi Number 1 $ 2.07
(jc 6 ) Alloy Number 2 41.01
(jc 7 ) Alloy Number 3 39.85
(x s ) SiC 6.71
These marginal costs indicate the amount of reduction in price per ton
that must occur before these materials can be included in the charge, i.e.,
Alloy Number 2 will be profitable to use only if its price per ton is re
duced by $41.01 per ton— from $260.00 to $218.99 per ton. These mar
ginal costs give management an indication of the penalties incurred for
deviating from the best solution. These can also be used very profitably
by the purchasing agent in his dealings with suppliers.
Additional marginal costs are provided which can be used to study the
effect of changing the chemical requirements of the charge. These marginal
costs for this problem are:
Element
Marginal Cost
Carbon
3.67
Silicon
1.98
Manganese
2.09
Chrome
5.09
which can be interpreted:
Carbon — If we raise the minimum carbon content in the charge by 1 per
cent this will increase the charge cost by $3.67 per ton. An increase of .1 percent
will increase the charge costs by $.37 per ton. Also a decrease of 1 percent in
the minimum carbon content will decrease the charge cost by $3.67 per ton,
A similar interpretation can be developed for each of the other elements.
The analysis of the marginal cost provided by the linear programming
solution are often of more value than the specific least cost charge in
that they give management a quantitative measure of penalties and hence
a measure of operating flexibility.
THE PROGRAM IN A PRODUCTION FOUNDRY
Aside from the mathematics involved in applying linear programming,
many problems can be encountered in putting this into effect in a produc
tion foundry.
In the summer of 1958 a study was initiated in a production foundry
to determine the feasibility of applying linear programming to the cupola
charge problem. During the early stages of the study it became apparent
that the linear programming calculations (for the simplex method) pre
sented an enormous chore. Therefore, a high speed computer was used for
these calculations. This is not to say that these calculations cannot be
A LINEAR PROGRAMMING APPLICATION TO CUPOLA CHARGING 47
accomplished by hand. However a high speed computer can solve a prob
lem like the one described previously in several minutes whereas it would
require three to six hours to accomplish the same thing by hand.
In addition to the bulk of calculations other problems were encountered.
Initially, the solutions which were obtained were not valid. Some of this
was due to incorrect data, errors in the data, and omissions in the state
ment of the problem. In one case, a solution specified a relatively high
usage of a certain scrap steel. This was not acceptable because the supply
of this scrap steel was sufficient for only a few days of operation at the
rate of usage indicated by the solution. Since a solution was desired which
could be used throughout one production month, it was necessary to restrict
the permissible amount of this scrap steel in the charge so that the available
quantity would last for one month's operation.
In another case, a solution indicated sizable savings over the present
charge. However this solution was not acceptable to the metallurgist be
cause, while the low limit on one element was .65 percent, the metallurgist
felt that the nature of the cupola operation required a charge with closer to
.70 percent of this element. Since the linear programming solution usually
develops a charge with exactly minimum amounts of the various elements,
the data had to be revised so that in effect the minimum specification on
this element was .70 percent. This type of thing may happen with the
more expensive elements in a charge.
In still another case, the solution was invalid because the data ignored
the fact that one charge material was not fully recovered in the melt.
Neglecting losses in melting usually leads to an unusable solution.
Errors appeared in one solution because several charge materials were
grouped together and handled collectively as if they were one material.
This resulted in erroneous answers because the costs of each material as
well as the chemical analysis was slightly different. It was just enough dif
ferent that when the materials were handled individually, the linear pro
gramming solution used all of one and none of the other materials. Here
it became apparent that in any solution, all possible materials should be
considered individually as separate materials even though costs and anal
yses are nearly the same. These early solutions, even though invalid, were
useful in demonstrating the sensitivity of the linear programming analysis
and the need for a complete statement of the problem.
It is important to realize that anyone beginning an analysis of cupola
charging via linear programming should not expect perfect results the
first time. Indeed, it is only after several invalid solutions that one fully
appreciates the ultimate value of the analysis.
In the early stages of the study, relatively few possible charge materials
were considered. Many materials were omitted because at one time in the
48 PLANNING AND CONTROLLING PRODUCTION LEVELS
past they were evaluated and were not profitable to use at that time. How
ever, as cupola charge specifications change, and with the fluctuations in
material costs from time to time, it was found that several of these materials
now could be profitably included in the charge. Hence, subsequent problems
have considered all possible charge materials. Here, it is necessary for us
to consider an important factor in understanding the application of linear
programming. Intuitive or inspection method may often yield the best
solution when relatively few charge materials are considered. These
methods, however, become increasingly less effective as the number of
materials to be considered increases. The larger the number of materials,
the more difficult it becomes to analyze the complex interaction of material
analyses and costs by cutandtry methods. Linear programming can
determine the lowest cost charge regardless of the number of charge ma
terials to be considered.
Some of the results of the early parts of the study were useful in indicat
ing the effects of a change in chemical specifications on charge cost. It was
found that for certain elements a very slight change (in hundredths of
percent) would show a marked increase or decrease in the cost and make
up of the least cost charge.
The initial study finally indicated that linear programming was feasible
and savings from $.15 to $.45 per ton might be realized. Therefore, in
the spring of 1959, it was decided to use linear programming for cupola
charging. It was decided to solve the problem twice per month in order to:
Determine the lowest cost charge considering all the materials available
for purchase;
Determine the lowest cost charge in terms of the materials on hand.
The first solution was developed from data prepared by the purchasing
agent and the metallurgist. The purchasing agent indicated what materials
were available, their delivered cost per ton, and available quantity. The
metallurgist specified the desired minimum chemical content of the charge,
the approximate chemical analyses of the charge materials, and any limita
tions on the use of certain charge materials. This solution was then used
as a guide by the purchasing agent in buying materials.
Later in the month the second solution was obtained. , Here the actual
costs, quantities, and chemical analyses of the materials on hand were
used to determine the least cost charge. This was used as a guide by the
metallurgist in charging the cupola.
In putting the linear programming on a production basis a form was
developed to facilitate data collection. Figure 1 illustrates the form that is
used for the linear programming analysis. Space is provided for the data
prepared by the purchasing agent, metallurgist, and the mathematician.
A LINEAR PROGRAMMING APPLICATION TO CUPOLA CHARGING
49
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50 PLANNING AND CONTROLLING PRODUCTION LEVELS
The analysis of certain materials remains relatively constant, so its analysis
is printed on the form. Each column in the upper half of the form repre
sents a charge material or space for one. The rows represent (from top to
bottom) :
1. The identification or name of the material.
2. Cost (including freight) — this may be cost per ton or cost per car load.
3. Code number — this identifies the material for the mathematician.
4. Total cost per net ton delivered — this is the numerical quantity used
in the linear programming analysis.
58. Four rows for the chemical content for carbon, silicon, manganese,
and chrome in the various materials and the charge.
You will note that plate (X25, X26, X27) and bundles (X28, X29,
X30) seem to have negative amounts of the various elements. During the
course of the study it was found that the use of plate and bundles in a
charge tends to reduce the recovery of certain of the elements in the final
melt. The negative numbers reflect the influence of these materials on
the recovery of the chemical elements. These corrections or modifications
were made after several months of operating experience with linear pro
gramming. It is interesting to note, here, that these corrections came to
light primarily due to the fact that the charges developed by linear pro
gramming always result in exactly the minimum chemical content. Without
these correction factors some charges resulted in melt which was below the
minimum specification on certain elements.
This form is started by the purchasing agent who fills in data concerning
the cost and availability of materials for charging the cupola. The form is
then sent to the metallurgist, who indicates the minimum chemical speci
fications, chemical analyses of the available charge materials, any limitations
on the use of charge materials, and the total tonnage projected for the
period.
The form is then sent to the mathematician who checks the data to
see that they are complete and converts the availability of the materials
in limited supply from tons to percent. The data are then ready for linear
programming.
The solution obtained with a high speed computer is recorded by the
mathematician on the data form. The solution is in two parts. First, the
percent of the various materials in the charge is noted in the Percent column.
The Dj column is used to record the marginal costs of the materials not
used in the charge. As mentioned earlier, linear programming not only
determines the best solution but it also provides information about less
thanbest solutions. The numbers (Dj) for those materials not used in
the charge indicate the reduction in price per ton required before these
A LINEAR PROGRAMMING APPLICATION TO CUPOLA CHARGING 51
materials could be profitably used in the charge. This is commonly called
a marginal cost. This, then, provides the purchasing agent with a bargain
ing point when talking with the scrap dealers. If prices can be obtained
which are lower than the original quoted prices, by the amount of the mar
ginal cost, then the purchasing agent purchases the lower priced material.
When this happens it becomes necessary to resolve the problem after the
materials are delivered in order to determine the cupola charge.
The second solution per month, if necessary, is handled by the metallur
gist and the mathematician since they are considering only those materials
available in inventory.
After four months of operation, a cost and savings analysis was made.
To be certain, it is difficult to exactly determine what might have been
done without linear programming. However, the analysis indicated savings
from $.10 to $1.40 per ton with the average at $.51 per ton for the
entire four month period. The cost of linear programming solutions must
be deducted from the foregoing. However, since the total cost per solution
is only about $25.00, this is quite negligible. Each problem would require
a day or more to solve by hand, but requires less than one minute on
an IBM 704 computer.
This application of linear programming has not only reduced cupola
charge costs, but it has opened the door to other operating cost reductions.
In fact, the other cost reductions in materials and operating procedures have
been several times the savings obtained by linear programming.
These savings have been realized in making malleable iron where we
are concerned with only a few chemical elements. In making aluminum
alloys, usually a greater number of elements with much closer specifications
must be considered. Here linear programming is even more valuable for
determining the least cost charge which satisfies all the specifications.
SUMMARY
In the past decade, technological progress has been made in mathematics
as well as in other fields of science. The results obtained by using linear
programming in cupola charging reflect the advantages of this new mathe
matical technique. Essentially, linear programming does the clerical task
that used to require a sizeable portion of the metallurgist's time and energy.
Linear programming has become an important new tool which depends
very heavily upon the close cooperation of metallurgist, purchasing agent,
and mathematician. This is a new addition to the many mathematical and
statistical tools available for analyzing foundry operating problems.
52 PLANNING AND CONTROLLING PRODUCTION LEVELS
References
(1) Metzger, R. W., Elementary Mathematical Programming, John Wiley &
Sons, 1958.
(2) Metzger, R. W., and Schwarzbek, R., "Least Cost Cupola Charging,"
presented before the General Motors Foundry Committee (Restricted
for General Motors distribution only).
Chapter 4
PRODUCTION PLANNING WITH
THE TRANSPORTATION MODEL
IV.
Mathematical Programming: Better
Information for Better Decision Making*
Alexander Henderson and Robert Schlaifer]
In recent years mathematicians have worked out a number of new pro
cedures which make it possible for management to solve a wide variety
of important company problems much faster, more easily, and more ac
curately than ever before. These procedures have sometimes been called
"linear programming." Actually, linear programming describes only one
group of them; "mathematical programming" is a more suitable title.
Mathematical programming is not just an improved way of getting certain
jobs done. It is in every sense a new way. It is new in the sense that
doubleentry bookkeeping was new in the Middle Ages, or that mechaniza
tion in the office was new earlier in this century, or that automation in
the plant is new today. Because mathematical programming is so new, the
gap between the scientist and the businessman — between the researcher
* From the Harvard Business Review, Vol. 32, No. 3 (1954), 73100. Reprinted by
permission of the Harvard Business Review.
t Authors' note: The authors wish to express their gratitude to Charles A. Bliss,
W. W. Cooper, and Abraham Charnes for their invaluable assistance in the prepara
tion of this article.
53
54 PLANNING AND CONTROLLING PRODUCTION LEVELS
and the user — has not yet been bridged. Mathematical programming has
made the news, but few businessmen really understand how it can be of use
in their own companies.
This article is an attempt to define mathematical programming for busi
nessmen, describe what it means in practice, and show exactly how to
use it to solve company problems. We have divided the article into four
sections :
Part I is addressed specifically to the top executive. Here are the salient
points about mathematical programming which the man who makes com
pany policy needs to know.
Part II is addressed to executives directly responsible for the organiza
tion and administration of operations where mathematical programming
could be used and to the specialists who actually work out the problems.
This part is based largely on case examples which are typical of the kinds
of problems that can be handled.
Part III shows management how to use mathematical programming as
a valuable planning tool. In many situations programming is the only
practical way of obtaining certain cost and profit information that is es
sential in developing marketing policy, balancing productive equipment,
making investment plans, and working out rational decisions on many other
kinds of shortrun and longrun problems.
In addition, to be used in connection with Part II, there is an appendix
providing actual instructions for working through the most frequently
useful, quick procedure for solving a common class of business problems.
PART I. BASIC PRINCIPLES
Production men usually have very little trouble in choosing which
machine tool to use for a given operation when there is free time available
on every tool in the plant. Traffic managers usually have little trouble in
choosing which shipping route to use when they are able to supply each
of their customers from the company's nearest plant. The manager of a
refinery usually has little trouble in deciding what products to make when
he has so much idle capacity that he can make all he can sell and more.
Except in the depths of depression, however, the problems facing man
agement are usually not this simple. Any decision regarding any one
problem affects not only that problem but many others as well. If an opera
tion is assigned to the most suitable machine tool, some other operation
MATHEMATICAL PROGRAMMING INFORMATION FOR DECISION MAKING 55
on some other part will have to be performed on some other, less suitable
tool. If Customer A is supplied from the nearest plant, that plant will not
have sufficient capacity to supply Customer B, who also is closer to that
plant than to any other. If the refinery manager makes all the 80octane
gasoline he can sell, he will not have capacity to satisfy the demand for
90octane gasoline.
BUSINESS PROGRAMS
The general nature of all these problems is the same. A group of limited
resources must be shared among a number of competing demands, and all
decisions are "interlocking" because they all have to be made under the
common set of fixed limits. In part, the limits are set by machinetool
capacity, plant capacity, raw materials, storage space, working capital, or
any of the innumerable hard facts which prevent management from doing
exactly as it pleases. In part, they are set by policies established by manage
ment itself.
When there are only a few possible courses of action — for example, when
a company with only two plants wants to supply three or four customers
at the lowest possible freight cost — any competent scheduler can quickly
find the right answer. However, when the number of variables becomes
larger — when a company has a dozen factories and 200 or 300 customers
scattered all over the country — the man with the job of finding the best
shipping pattern may well spend many days only to end up with a
frustrated feeling; though he thinks he is close to the right answer, he is
not at all sure that he has it. What is worse, he does not even know how
far off he is, or whether it is worth spending still more time trying to im
prove his schedule. The production manager who has 20 or 30 different
products to put through a machine shop containing 40 or 50 different
machine tools may well give up as soon as he has found any schedule that
will get out the required production, without even worrying whether some
other schedule would get out the same product at a lower cost.
Under these conditions business may incur serious unnecessary costs
because the best program is not discovered. Another kind of cost is often
even more serious. The few direct tests which have been made so far
show that intelligent and experienced men on the job often (though
by no means always) come very close to the "best possible" solution of
problems of this sort. But since problems of such complexity can almost
never be handled by clerical personnel, even these good cutandtry solu
tions are unsatisfactory because they take up a substantial amount of the
56 PLANNING AND CONTROLLING PRODUCTION LEVELS
time of supervisory employees or even of executives.
The time of such men is the one thing that management cannot readily
buy on the market. If it is all used up just in getting the necessary informa
tion, there is nothing left for the next step, making sound decisions. Often
this produces a sort of inertia against any change in the status quo; it is
so hard to find out the cost or profit implications of a proposed change or
series of changes that management simply gives up and lets existing
schedules and programs stand unchanged. Conversely, if better informa
tion were available more easily, management would be less tempted to
drop important questions without investigation or could make better de
cisions as a result of investigation.
A ROUTINE PROCEDURE
Many of these complex and timeconsuming problems can in fact be
solved today by mathematical programming. The purely routine procedures
of which it is comprised can be safely entrusted to clerical personnel or
to a mechanical computer. Such procedures have already been successfully
applied to practical business problems, some of which will be described
in the course of this article.
The word mathematical may be misleading. Actually the procedures go
about solving problems in much the same way as the experienced man on
the job. When such a man is faced by a problem with many interlocking
aspects, he usually starts by finding a program that meets the minimum
requirements regardless of cost or profit, and then tries out, one by one,
various changes in this program that may reduce the cost or increase the
profit. His skill and experience are required for two reasons: (a) to perceive
the desirable changes and (b) to follow through the repercussions of a
single change on all parts of the program.
What "mathematical" programming does is to reduce the whole procedure
to a simple, definite routine. There is a rule for finding a program to start
with, there is a rule for finding the successive changes that will increase
the profits or lower the costs, and there is a rule for following through all
the repercussions of each change. What is more, it is absolutely certain
that if these rules are followed, they will lead to the best possible program;
and it will be perfectly clear when the best possible program has been
found. It is because the procedure follows definite rules that it can be
taught to clerical personnel or handed over to automatic computers.
COST INFORMATION
Quick and inexpensive calculation of the best possible programs or
schedules under a particular set of circumstances is not the only benefit
MATHEMATICAL PROGRAMMING INFORMATION FOR DECISION MAKING
51
which management can obtain from this technique. The same complex
situation which makes it difficult to find the best possible schedule for
the entire operation makes it difficult to get useful cost information con
cerning details of the operation. When every operation in the shop can be
performed on the most suitable machine tool, the cost of any particular
operation can be obtained by the usual methods of cost accounting. But
if capacity is short, then the true cost of using a machine for one particular
operation depends in a very real sense on the excess costs incurred be
cause some other part has to be put on a less suitable machine. To il
lustrate further:
If the production of 80octane gasoline is carried to a point where less 90
octane can be produced than can be sold, the profits which failed to be made on
90octane must certainly be kept in mind when looking at the stated profits on
the 80octane.
When a company is supplying some (but not all) of its eastern customers by
bringing in supplies from the West Coast, additional cost will be incurred by
giving one of these customers quick delivery from a nearby plant, even though
the actual freight rate from the nearby plant is lower than the rate from the
West Coast.
Any time that the programming procedure will solve the basic problem
of determining the most profitable overall schedule, it will also produce
usable cost information on parts of the whole operation. In many cases
this information may be even more valuable than the basic schedule. It
can help management decide where to expand plant capacity, where to
push sales, and where to expend less effort, or what sorts of machine
tools to buy on a limited capital budget. In the long run, sound decisions
in matters of this sort will pay off much more substantially than the choice
of the best shipping program in a single season or the best assignment of
machine tools for a single month's production.
LIMITATIONS
Mathematical programming is not a patented cureall which the business
man can buy for a fixed price and put into operation with no further
thought. The principal limitations of the technique today lie in three
areas :
1. Cost or revenue proportional to volume — Problemsolving procedures have
been well developed only for problems where the cost incurred or revenue pro
duced by every possible activity is strictly proportional to the volume of that
activity; these are the procedures that belong under the somewhat misleading
title of linear programming. This limitation, however, is not so serious as it
58 PLANNING AND CONTROLLING PRODUCTION LEVELS
seems. Problems involving nonproportional costs or revenues can often be
handled by linear programming through the use of special devices or by suitable
approximations, and research is progressing on the development of procedures
which will handle some of these problems directly.
2. Arithmetic capacity — Even when the procedure for solving a problem is
perfectly well known, the solution may involve such a sheer quantity of arith
metic that it is beyond the capacity even of electronic computing machinery.
However, the problem can sometimes be set up more simply so that solution
is practical. For instance, careful analysis may show that the really essential
variables are relatively few in number, or that the problem may be split into
parts of manageable size.
3. Scheduling problems — A third limitation is often the most serious, particu
larly in the assignment of machine tools. So far very little has been accomplished
toward the solution of scheduling problems, where certain operations must be
performed before or after other operations. Mathematical programming can in
dicate, within the limits of available tool capacity, which operations should be
performed on which tools, but the arrangement of these operations in the proper
sequence must usually be handled as a separate problem. Again, however, re
search is attempting to find procedures which will reduce even this problem to
a straightforward routine, and some progress in this direction has already been
made.
APPLICATION
In Part II of this article we describe a series of cases which should
suggest to the reader the sort of problems where mathematical program
ming can be of use in his own business. Included are both actual cases and
hypothetical examples. The hypothetical examples are purposely made
so simple that they could be solved without the use of these procedures; in
this way the reader can better see the essential nature of the analysis
which programming will accomplish in more complex problems.
Top executives may want to turn a detailed reading of this section over
to specialists, but they will find the major points as set forth below of
practical interest. Very briefly, the discussion of case examples will show
that mathematical programming can be used to decide:
1 . Where to ship — Here the problem is to find the shipping program that will
give the lowest freight costs. It has been demonstrated by the H. J. Heinz Com
pany that linear programming can save thousands of dollars on a single schedul
ing problem alone. By virtue of its greater ease and accuracy, linear program
ming has also enabled the company to schedule on a monthly rather than
quarterly basis, thus taking advantage of new information as soon as it becomes
MATHEMATICAL PROGRAMMING INFORMATION FOR DECISION MAKING 59
available.
2. Where to ship and where to produce — A complete program to determine
the most economical program of production or procurement and freight costs
can be developed so quickly and inexpensively that every possible alternative
can be taken into account without throwing a heavy burden on senior personnel.
3. Where to ship, where to produce, and where to sell — Here the problem is
further complicated. Such factors as a management policy regarding minimum
supplies for dealers and a varying price schedule should and can be taken into
account.
4. What the most profitable combination of price and volume is — At present
mathematical programming can provide the answers only under certain condi
tions, but progress is being made in broadening its applicability.
5. What products to make — Problems that can be solved range from the most
economical use of scarce raw materials to the most profitable mix in gasoline
blending. If automatic computers are necessary because of the sheer bulk of
arithmetic, the small or mediumsize firm can turn to a central service bureau;
the company does not have to be so large that it can afford its own computers.
6. What products to make and what processes to use — This problem arises
when machine capacity is limited. Here mathematical programming may produce
surprising results. For example, a certain amount of idle time on one machine
may be necessary for the greatest production. Without mathematical program
ming, there is a real danger that personnel will use every machine all the time to
satisfy management pressure, and thus defeat the company's real objective.
7. How to get lowest cost production — Here the problem is to determine the
most economical production when the company can produce all it can sell. In
these days of growing costconsciousness, mathematical programming may be
come one of management's really valuable costreduction tools.
The businessman who recognizes or suspects that he has a problem which
can be solved by mathematical programming will usually have to consult
with specialists to learn how to use the technique. But an even greater
responsibility will reman with the businessman himself. Like the introduc
tion of a variable overhead budget, each application of mathematical pro
gramming will require careful study of the particular circumstances and
problems of the company involved; and, once installed, the technique will
pay off only in proportion to the understanding with which management
makes use of it.
PART II. EXAMPLES OF OPERATION
The case examples to be presented here illustrate some of the uses of
mathematical programming. Although limited in number, the examples are
60 PLANNING AND CONTROLLING PRODUCTION LEVELS
so arranged that the reader who follows them through in order should gain
an understanding of the situations in which mathematical programming can
and cannot be helpful and of how to set up any problem for accurate
solution. The exhibits accompanying the text set forth the mathematical
solution of the problems posed in the cases, while the appendix gives
specific directions on how to work through a procedure for handling some
of the problems that may arise in the reader's own business.
WHERE TO SHIP
As our first example of the uses of mathematical programming, let us
look at a case where the technique is currently in use as a routine operating
procedure in an actual company:
The H. J. Heinz Company manufactures ketchup in half a dozen plants
scattered across the United States from New Jersey to California and distributes
this ketchup from about 70 warehouses located in all parts of the country.
In 1953 the company was in the fortunate position of being able to sell all it
could produce, and supplies were allocated to warehouses in a total amount
exactly equal to the total capacity of the plants. Management wished to supply
these requirements at the lowest possible cost of freight; speed of shipment was
not important. However, capacity in the West exceeded requirements in that
part of the country, while the reverse was true in the East; for this reason a
considerable tonnage had to be shipped from western plants to the East. In other
words, the cost of freight could not be minimized by simply supplying each
warehouse from the nearest plant.
SIMPLEST PROBLEM
This problem can immediately be recognized as a problem of program
ming because its essence is the minimization of cost subject to a fixed set of
plant capacities and warehouse requirements. It can be handled by linear
programming because the freight bill for shipments between any two points
will be proportional to the quantity shipped. (The quantities involved are
large enough so that virtually everything will move at carload rates under
any shipping program which might be chosen.)
This is, in fact, the simplest possible kind of problem that can be solved
by this method. Certain complexities which make solution by trial and
error considerably more difficult than usual — in particular, the existence
of watercompetitive rates, which make it practical to send California
ketchup all the way to the East Coast — add no real difficulty to the solu
tion by linear programming. Given the list of plant capacities and warehouse
MATHEMATICAL PROGRAMMING INFORMATION FOR DECISION MAKING 61
requirements, plus a table of freight rates from every plant to every ware
house, one man with no equipment other than pencil and paper solved
this problem for the first time in about 12 hours. After H. J. Heinz had
adopted the method for regular use and clerks had been trained to become
thoroughly familiar with the routine for this particular problem, the time
required to develop a shipping program was considerably reduced.
The actual data of this problem have not been released by the company,
but a fair representation of its magnitude is given by the similar but hypo
thetical examples of Exhibits I and II, which show the data and solution of
a problem of supplying 20 warehouses from 12 plants.
Exhibit I shows the basic data: the body of the table gives the freight
rates, while the daily capacities of the plants and daily requirements of
the warehouses are in the margins. For example, Factory III, with a
capacity of 3,000 cwt. per day, can supply Warehouse G, with require
ments of 940 cwt. per day, at a freight cost of 7 cents per cwt.
Any reader who wishes to try his hand will quickly find that without
a systematic procedure a great deal of work would be required to find a
shipping program which would come reasonably close to satisfying these
requirements and capacities at the lowest possible cost. But with the
use of linear programing the problem is even easier than the Heinz problem.
Exhibit II gives the lowestcost distribution program. For example,
Warehouse K is to get 700 cwt. per day from Factory I and 3,000 cwt.
per day from Factory III. On the other hand, Factory III ships nothing to
Warehouse A, although Exhibit I shows that Factory III could ship at
less expense to this warehouse than to any other. (The "row values" and
"column values" are cost information, the meaning of which is explained
on p. 94.)
62
PLANNING AND CONTROLLING PRODUCTION LEVELS
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64 PLANNING AND CONTROLLING PRODUCTION LEVELS
ADVANTAGES GAINED
One of the most important advantages gained by the H. J. Heinz Com
pany from the introduction of linear programming was relief of the senior
members of the distribution department from the burden of preparing
shipping programs. Previously the quarterly preparation of the program
took a substantial amount of their time; now they pay only as much atten
tion to this problem as they believe necessary to keep the feel of the
situation, while the detailed development of the program has been handed
over to clerks. Freed from the burden of working out what is after all
only glorified arithmetic, they have this much more time to devote to
matters which really require their experience and judgment.
An equally important gain, in the opinion of these officials themselves,
is the peace of mind which results from being sure that the program is
the lowestcost program possible.
The direct dollarsandcents saving in the company's freight bill was
large enough by itself to make the use of this technique very much worth
while. The first shipping program produced by linear programming gave a
projected semiannual freight cost several thousand dollars less than did a
program prepared by the company's previous methods, and this com
parison is far from giving a full measure of the actual freight savings
to be anticipated.
Shipping schedules rest on estimates which are continuously subject to
revision. The capacity figures in part represent actual stocks on hand at
the plants, but in part they are based on estimates of future tomato crops;
and the figures for requirements depend almost wholly on estimates of
future sales. The fact that schedules are now quickly and accurately pre
pared by clerks has enabled the company to reschedule monthly rather
than quarterly, thus making much better use of new information on crops
and sales as it becomes available.
Furthermore, the risk of backhauling is very much reduced under the
new system. It had always been company practice early in the season to
hold "reserves" in regions of surplus production, in order to avoid the
danger of shipping so much out of these regions that it became necessary
to ship back into them when production and sales estimates were revised.
In fact, these reserves were largely accidental leftovers: when it became
really difficult to assign the last part of a factory's production, this re
mainder was called the reserve. Now the company can look at past history
and decide in advance what reserve should be held at each factory and
can set up its program to suit this estimate exactly. Since the schedule is
revised each month, these reserves can be altered in the light of current
information until they are finally reduced to nothing just before the new
pack starts at the factory in question.
MATHEMATICAL PROGRAM MING INFORMATION FOR DECISION MAKING 65
SIMILAR PROBLEMS
Many important problems of this same character unquestionably are
prevalent in business. One such case, for instance, would be that of a news
print producer who supplies about 200 customers all over the United
States from 6 factories scattered over the width of Canada. 1
Similar problems arise where the cost of transportation is measured in
time rather than in money. In fact, the first efforts to solve problems of
this sort systematically were made during World War II in order to minimize
the time spent by ships in ballast. Specified cargo had to be moved from
specified origins to specified destination; there was usually no return
cargo, and the problem was to decide to which port the ship should be
sent in ballast to pick up its next cargo. An obviously similar problem
is the routing of empty freight cars, 2 and a trucker operating on a nation
wide scale might face the same problem with empty trucks.
WHERE TO PRODUCE
When ketchup shipments were programmed for the H. J. Heinz Com
pany, factory capacities and warehouse requirements were fixed before the
shipping program was worked out, and the only cost which could be
reduced by programming was the cost of freight. Since management had
decided in advance how much to produce at each plant, all production
costs were "fixed" so far as the programming problem was concerned.
The same company faces a different problem in connection with another
product, which is also produced in a number of plants and shipped to a
number of warehouses. In this case, the capacity of the plants exceeds the
requirements of the warehouses. The cost of production varies from one
plant to another, and the problem is thus one of satisfying the requirements
at the least total cost. It is as important to reduce the cost of production
(by producing in the right place) as it is to reduce the cost of freight (by
supplying from the right place.) In other words, management must now
decide two questions instead of one: (a) How much is each factory to
produce? (b) Which warehouses should be supplied by which factories?
It is tempting to try to solve these two problems one at a time and
thus simplify the job, but in general it will not be possible to get the
lowest total cost by first deciding where to produce and then deciding where
to ship. It is obviously better to produce in a highcost plant if the additional
cost can be more than recovered through savings in freight.
METHOD OF ATTACK
This double problem can be handled by linear programming if we may
1 R. Dorfman, "Mathematical, or 'Linear' Programming," American Economic
Review, Dec. 1953, p. 797.
2 Cf. Railway Age, April 20, 1953, pp. 7374.
66
PLANNING AND CONTROLLING PRODUCTION LEVELS
assume (as businessmen usually do) that the cost of production at any
one plant is the sum of a "fixed" cost independent of volume and a
"variable" cost proportional to volume in total but fixed per unit, and if
these costs are known. The variable cost is handled directly by the linear
programming procedure, while the fixed part is handled by a method which
will be explained later.
Actually, the problem can be much more complicated and still lend itself
to solution by linear programming. For example, we can bring in the possi
bility of using overtime, or of buying raw materials at one price up to a
certain quantity and at another price beyond that quantity. (Although
there is no longer a constant proportion between production and variable
cost, we can restore proportionality by a device described in the Appendix,
p. 101.)
Exhibit III shows the cost information needed to solve a hypothetical
example of this sort. It is assumed that there are only four plants and four
warehouses, but any number could be brought into the problem.
In our first approximation (which we shall modify later) we shall assume
that no plant will be closed down entirely and, therefore, that "fixed costs"
are really fixed and can be left out of the picture. Like Exhibit I, Exhibit
III shows the freight rates from each plant to each warehouse, the available
daily capacity at each plant, and the daily requirements of each warehouse;
it also shows the "variable" (fixedperunit) cost of normal production at
each plant and the additional perunit cost of overtime production. The
total capacity is greater than the total requirements even if the factories
work only normal time.
EXHIBIT III. COST INFORMATION FOR DOUBLE PROBLEM
A — Warehouse Requirements {tons per day)
A B C D
Warehouse
Requirements
Factory
90
140
75
Factory Capacities {tons per day)
I II III
Normal capacity 70 130 180
Additional capacity on overtime 25 40 60
Factory
C — Variable Costs {per ton)
I II III
100
IV
110
30
IV
Total
405
Total
490
155
Normal production cost
$30
$36
$24
$30
Overtime premium
15
18
12
15
Freight rates to:
Warehouse A
$14
$ 9
$21
$18
B
20
14
27
24
C
18
12
29
20
D
19
15
27
23
MATHEMATICAL PROGRAMMING INFORMATION FOR DECISION MAKING
67
On the basis of these data, the lowestcost solution is given by Part A of
Exhibit IV. It is scarcely surprising that this solution calls for no use of
overtime. So long as fixed costs are taken as really fixed, it turns out that it
is best to use the entire normal capacity of Factories I, II, and III, and to
use 25 tons of Factory IV's normal capacity of 110 tons per day. The re
maining 85 tons of normal capacity at IV are left unused. The total variable
cost under this schedule (freight cost plus variable production cost) will be
$19,720 per day.
EXHIBIT
IV.
LOWEST COST DISTRIBUTION
PROGRAM
{Daily shipments in tons from factory
to warehouse)
A — With All Four Factories Open
Factory
I II
III
IV
Total
Warehouse A
90
90
B
80
60
140
C
50
25
75
D
70
30
100
Idle normal capacity
85
85
Total
70 130
180
110
490
B—With Factory I Closed
Factory
II
III
IV
Total
Warehouse A
90
90
B
130
10
140
C
75
75
D
80
20
100
Idle normal capacity
15
15
Total
130
180
110
420
Factory
C—With Factory IV Closed
I II III
Warehouse A
90
B
55
85
C
75
D
70
30
Total
70
130
205
Total
90
140
75
100
405
FINAL DETERMINATION
Presented with this result, management would certainly ask whether it is
sensible to keep all four factories open when one of them is being left about
80% idle. Even without incurring overtime, Factory I, the smallest plant,
could be closed and the load redistributed among the other plants. If this is
done, the lowestcost distribution of the requirements among Factories II,
III, and IV is that given by Part B of Exhibit IV. Under this program the
total variable cost would be $19,950 per day, or $230 per day more than
68 PLANNING AND CONTROLLING PRODUCTION LEVELS
under the program of Exhibit IV, A, which depended on the use of all four
plants. If more than $230 per day of fixed costs can be saved by closing
down Factory I completely, it will pay to do so; otherwise it will not.
It might be still better, however, to close down some plant other than
Factory I even at the cost of a certain amount of overtime. In particular, a
very little overtime production (25 tons per day) would make it possible to
close Factory IV. A person asked to look into this possibility might reason
as follows: Under the shipping schedule of Exhibit IV, A, the only use of
Factory IV's capacity is to supply 25 tons per day to Warehouse C. Look
ing at Exhibit III for a replacement for this supply, he would get the fol
lowing information on costs per ton :
Factory
Normal cost
of production
Overtime
premium
Freight to
Warehouse C
Total
I
II
III
$30
36
24
$15
18
12
$18
12
29
$63
66
65
Apparently the cheapest way of using overtime, if it is to be used at all,
would be to produce the needed 25 tons per day at Factory I and ship them
to Warehouse C at a total variable cost of $63 per ton. Under the program
of Exhibit IV, A, with all plants in use, Warehouse C was supplied from
Factory IV at a total variable cost of $30 for production plus $20 for
freight, or a total of $50 per ton. The change would thus seem to add a
total of $325 per day (25 tons times $13 per ton which is the difference
between $63 and $50 per ton).
But, in fact, closing Factory IV need not add this much to the cost of the
program. If we take Factory IV out of the picture and then program to find
the best possible distribution of the output of the remaining plants, we dis
cover that the program of Part C of Exhibit IV satisfies all requirements at
a total variable cost of $19,995 per day, or only $275 per day more than
with all plants in use. The overtime is performed by Factory III, which does
not supply Warehouse C at all.
DIFFICULTIES AVOIDED
This last result deserves the reader's attention. Once a change was made
in a single part of the program, the best adjustment was a general readjust
ment of the entire program. But such a general readjustment is impractical
unless complete programs can be developed quickly and at a reasonable
cost. It is rarely clear in advance whether the work will prove profitable,
and management does not want to throw a heavy burden of recalculation on
senior personnel every time a minor change is made. Mathematical pro
gramming avoids these difficulties. Even minor changes in the data can be
MATHEMATICAL PROGRAMMING INFORMATION FOR DECISION MAKING 69
made freely despite the fact that complete recalculations of the program are
required, because the work can be done quickly and accurately by clerks or
machines.
We can proceed to compute the lowest possible cost of supplying the
requirements with Factory II or Factory III closed down completely. We
can then summarize the results for all alternatives like this:
Total freight plus variable production cost
All four factories in use $19,720
Factory I closed, no overtime 19,950
Factory II closed, overtime at Factory HI 20,515
Factory III closed, overtime at Factories I, II, and IV 21,445
Factory IV closed, overtime at Factory III 19,995
Management now has the information on variable costs which it needs in
order to choose rationally among three alternatives : ( 1 ) operating all four
plants with a large amount of idle normal capacity; (2) shutting down
Factory I and still having a little idle normal capacity; (3) shutting down
Factory II, III, or IV and incurring overtime. Its choice will depend in
part on the extent to which fixed costs can be eliminated when a particular
plant is completely closed; it may depend even more on company nolicies
regarding community relations or some other nonfinancial consideration.
Mathematical programming cannot replace judgment, but it can supply some
of the factual information which management needs in order to make
judgments.
RELATED PROBLEMS
Problems of this general type are met in purchasing as well as in pro
ducing and selling. A company which buys a standard raw material at many
different geographical locations and ships it to a number of scattered plants
for processing will wish to minimize the total cost of purchase plus freight;
here the solution can be obtained in exactly the same way as just discussed.
The Department of Defense is reported to have made substantial savings
by using linear programming to decide where to buy and where to send
certain standard articles which it obtains from a large number of suppliers
for direct shipment to military installations.
WHERE TO SELL
In our first case, we considered a situation where management had fixed
the sales at each warehouse and the production at each plant before using
programming to work out the best way of shipping from plant to warehouse.
In the second example, management had fixed the sales at each warehouse in
advance, but had left the decision on where and how much to produce to
70 PLANNING AND CONTROLLING PRODUCTION LEVELS
be made as a part of the program. Let us now consider a case in which sales
are not fixed in advance, and management wants to determine where to sell,
as well as where to produce and where to ship, in order to give the greatest
possible profits.
Such a problem often arises when sales would exceed a company's
capacity to produce unless demand were retarded by higher prices, yet
management does not wish to raise prices because of the longrun competi
tive situation. Under these circumstances some system of allocating the
product to branch warehouses in the different market areas (or to individual
customers) will be necessary. One way of doing this is simply to sell
wherever the greatest shortrun profits can be made. Often, however, man
agement will not want to take an exclusively shortrun view and will want
to provide each warehouse or customer with at least a certain minimum
supply, with only the remainder over and above these minimum allocations
being disposed of with a view to maximum shortrun profits.
One additional complication will often be present in real problems of this
sort. The selling price of the product may not be uniform nationally, but
may vary from place to place or from customer to customer. In addition,
there may well be present the complication we dealt with in the last ex
ample: it may be desirable to have some plants working overtime while
others are working at only a part of their normal capacity or are even closed
down entirely.
Thus a production and distribution program must be prepared which
answers all the following questions in such a way as to give the greatest
possible profits, subject to the requirement of supplying certain warehouses
with at least a specified allocation of product:
( 1 ) How much shall be produced at each plant?
(2) How much, if any, above the predetermined minimum shall be delivered
to each warehouse?
(3) The above questions being answered, which plants shall supply which
warehouses?
As in the previous example, all three questions must be answered simul
taneously; it is not possible to work them out one by one. The problem can
still be handled by linear programming, however, despite the additional
complications which have entered the picture; in fact, it is no harder to
solve than the previous problem. The only difference is that we now look
directly at the profit resulting from supplying a particular warehouse from
a particular plant, rather than looking at the costs involved. We shall not
even work out an example, since the solution would appear in the same
form as Exhibit IV of the previous case, while the required data would look
the same as Exhibit III with the addition of the selling price at each ware
house.
MATHEMATICAL PROGRAMMING INFORMATION FOR DECISION MAKING 71
PRICE, VOLUME, AND PROFIT
In all the previous examples it was assumed that management had set
selling prices before the production and distribution program was worked
out. The quantity to be produced and shipped followed from the predeter
mined prices. This is certainly a common situation, but it is also very
common for management to want to consider the effect of prices on volume
before prices are set. This means, of course, that sales volume must be
forecast at each of a variety of possible prices, and we assume that such
forecasts have been made separately for each of the branch warehouses of
our previous examples.
Under these conditions the problem can no longer be handled directly by
linear programming, since the margin, or difference between the selling price
at a particular warehouse and the variable cost of producing at a particular
plant and shipping to that warehouse, is no longer in a constant ratio to the
quantity produced and sold. As quantities go up, prices go down, and the
ratio of total margin to quantity sold declines. Even so, we can still use
linear programming to solve the problem quickly, accurately, and cheaply if
there is to be a single national selling price. We can compute the best pro
gram for each proposed price, determine the total profits for each program,
and select the most profitable alternative.
However, linear programming becomes virtually impossible if prices can
vary from place to place and management wishes to set each local price in
such a way as to obtain the greatest total profits. Even if there are only ten
distribution points for which pricequantity forecasts have to be considered,
and even if each branch manager submits forecasts for only five different
prices, we would have to compute nearly 10 million different programs and
then select the most profitable one.
In practical cases it will often prove possible with a reasonable amount
of calculation to find a program which is probably the best program or very
close to it, but in general the solution of this problem of mathematical pro
gramming, like many others, depends on further research to develop meth
ods for attacking nonlinear problems directly. As mentioned, progress in this
direction is already being made.
WHAT AND HOW TO PRODUCE
All the cases discussed so far have involved problems of where (as well
as how much) to buy, sell, produce, and ship. Mathematical programming
can be of equal use in deciding what and how to produce in order to
maximize profits or minimize costs in the face of shortages of raw materials,
machine tools, or other productive resources. Some problems of this kind
may be solved by clerks using procedures such as those previously discussed;
72
PLANNING AND CONTROLLING PRODUCTION LEVELS
others, however, may require new procedures and automatic computing
equipment.
A representative problem in the first category is the following one, which
involves the selective use of scarce raw materials:
A manufacturer produces four products, A, B, C, and D, from a single raw
material which can be bought in three different grades, I, II, and III. The cost
of processing and the quantity of material required for one ton of end product
vary according to the product and the grade of material used, as shown in
Exhibit V.
EXHIBIT V. COSTS, AVAILABILITIES, AND
PRICES
Grade
A — Yields and Processing Costs
I II
III
Product
Tons of material per
ton of product
A
B
C
D
1.20 1.80
1.50 2.25
1.50 2.25
1.80 2.70
2.00
2.50
2.50
3.00
Processing cost per ton of product
A
B
C
D
$18 $30
30 60
57 63
54 81
B — Material Cost and Availability
$ 42
69
66
126
Grade
I II
III
Normal price per ton $48 $24
Quantity available at
normal price (tons) 100 150
Premium price pef ton $72 $36
Quantity available at
premium price (tons) 100 150
$18
250
$24
400
C — Product Prices and Sales Potentials
Product
A B
C
D
Price per
Potential
ton $96 $150
sales (tons) 200 100
$135
160
$171
50
If unlimited supplies of each grade of material were available at a fixed
market price, each product would be made from the grade for which the total
purchasingplusprocessing cost was the smallest; but the amount of each grade
obtainable at the "normal" price is limited as shown in the exhibit. Additional
quantities of any grade can be obtained, but only at the premium shown.
The products are sold f.o.b. the manufacturer's single plant; the selling prices
have already been set and are shown in the exhibit, together with the sales
department's forecasts of the amount of each product which can be sold at
these prices.
MATHEMATICAL PROGRAMMING INFORMATION FOR DECISION MAKING 73
The problem, then, is to determine what products to make and how much
of each, and how to make them — in other words, which grade of material
to use for which products. The solution is shown in Exhibit VI.
EXHIBIT VI. MOST PROFITABLE PRODUCTION PROGRAM
Tons of pi
'oduct
Produc
tion
Tons of material used
Sales
Product potential
Grade Grade Grade
I II III
A 200
B 100
C 160
D 50
200
100
160
210 167
100 83
400
Total material usage
100 210 650
Bought at normal price
Bought at premium price
100 150 250
60 400
USE OF COMPUTERS
It will be remembered that, in discussing the use of mathematical pro
gramming by the H. J. Heinz Company, we emphasized the fact that ship
ping programs are produced by a clerk with nothing but paper and pencil in
a very reasonable amount of time. This is true even though the existence of
about 6 plants and 70 warehouses makes it necessary to choose 75 routes
for actual use from the 420 possible routes which might be used. This ease
of solution, even in cases where a very large number of variables is involved,
applies to the selective use of raw materials just discussed as well as to the
other problems taken up in earlier sections. They are all problems which
can be solved by what is known as the "transportationproblem procedure."
By contrast, other problems usually require the use of highspeed com
puting machinery. They are problems requiring the use of what might be
called the "general procedure." While the mathematics involved here is at
the level of gradeschool arithmetic, the sheer bulk of arithmetic required
is very much greater than under the transportation procedure. This means
that, unless a skilled mathematician finds some way of simplifying a par
ticular problem, it will be impossible for clerks to obtain a solution by
hand in a reasonable amount of time when the number of variables is such
as will be encountered in most practical situations.
Whether a given problem can be solved by the transportationproblem
procedure or will require the use of the general procedure does not depend
on whether the problem actually involves transportation or not, but rather
on the form of the data. The raw material problem discussed just above,
74 PLANNING AND CONTROLLING PRODUCTION LEVELS
for example, could be solved as a transportation problem because any
product would require 50% more material if Grade II was used instead of
Grade I, or 67% more if Grade III was used instead of Grade I. But if the
inferiority of yield of the lower grades had varied depending on the par
ticular end product, it would have been necessary to use the general pro
cedure.
The fact that the general procedure usually requires an automatic com
puter by no means implies that this procedure can be profitably applied
only by very large firms with computers of their own. Fortunately, all prob
lems which call for the use of this procedure are mathematically the same,
even though the physical and economic meaning of each problem may be
completely different. And since they are mathematically the same, a ma
chine at a central service bureau can be coded once and for all to carry out
the general procedure for any problem up to a certain size. The machine
can then be used to solve the varying problems of many different companies
promptly and inexpensively. Such a service can already be purchased from
at least one source by the hour, and the time required to solve a problem is
usually surprisingly short.
MOST PROFITABLE BLEND
Now let us turn to a case requiring the use of the general procedure:
Gasoline sold as an automobile or aviation fuel is ordinarily not the product
of a single refining process but a blend of various refinery products with a
certain amount of tetraethyl lead added. To a certain extent each of the various
constituents requires peculiar refining facilities. Consequently, the management
of a refinery may well be faced with the following problem: given a limited
daily supply of each of various constituents, into what endproduct fuels should
they be blended to bring in the maximum profits? The problem is made addi
tionally complicated by the fact that there is no single "recipe" for any par
ticular end product. In general, the end product may be blended in any of a
large number of different ways, provided only that certain performance specifi
cations are met.
This is clearly a problem of programming, both because the use of a given
constituent in one end product means that less is available for use in
another, and also because the use of one constituent to produce a given
kind of performance in a particular end product means that less of other
constituents is needed to produce that performance in the end product. But
is the problem linear? We must look a little more closely at the relation
between the characteristics of the constituents and the characteristics of the
resulting blends:
MATHEMATICAL PROGRAMMING INFORMATION FOR DECISION MAKING 15
The two most important measures of the performance characteristics of a
gasoline fuel are its performance number (PN), which is a development of the
octane number and describes antiknock properties, and its vapor pressure
(RVP), which indicates the volatility of the fuel. In the case of most highgrade
aviation gasolines there are actually two PN's specified: the 1c PN, which
applies to lean mixture, and the 3c PN, which applies to rich mixture. Each of
the various constituents has its own RVP and PN.
The PN and RVP required in the end product are produced by proper blend
ing of the constituents and by the addition of tetraethyl lead (TEL) to improve
the PN. The amount of TEL which can be used in any fuel is limited for various
reasons; and since TEL is often the cheapest way of obtaining the desired PN
(particularly in the case of aviation fuels), it is a common practice to use the
maximum permitted amount of this chemical.
It appears from the above that the problem will be linear provided that
the RVP and PN of any end product are simply weighted averages of the
RVP's and PN's of the various constituents (each PN being calculated for
the predetermined amount of TEL to be used in the end product). While
not perhaps strictly true as regards PN, this proposition is close enough to
the truth to serve as the basis for ordinary blending calculations. Therefore
the problem can be handled in a straightforward manner by linear pro
gramming.
A. Charnes, W. W. Cooper, and B. Mellon have applied linear program
ming to the choice of the most profitable mix in an actual refinery; and
although they were forced to simplify the problem somewhat in order to do
the computation with nothing but a desk calculator, the results of their
calculations were of considerable interest to the company's management. 3
(With modern computing equipment, of course, much more data could be
handled in much less time, and various large oil companies are currently
trying out the use of such equipment for this purpose.)
The figures which Charnes, Cooper, and Mellon present to show the
nature of the calculations, and which we use below, are of course largely
disguised:
The refinery in question is considered as having available fixed daily supplies
of one grade of each of four blending constituents: alkylate, catalyticcracked
gasoline, straightrun gasoline, and isopentane. The quantities available and the
performance specifications are shown in Exhibit VII. These constituents can be
blended into any of three different aviation gasolines, A, B, or C, the specifica
tions and selling prices of which are also shown in Exhibit VII.
3 A. Charnes, W. W. Cooper, and B. Mellon, "Blending Aviation Gasolines — A
Study in Programming Interdependent Activities in an Integrated Oil Company,"
Econometrica, April 1952, p. 135.
76
PLANNING AND CONTROLLING PRODUCTION LEVELS
EXHIBIT VII.
QUANTITIES AVAILABLE AND PERFORMANCE
SPECIFICATIONS
Product
A — Product Specifications
Maximum
TEL
Cost of
Maxi
Mini
Mini
cc. per
Price
TEL per
mum
mum
mum
gal of
per bbl.
bbl. of
RVP
1cPN
3cPN
product
of product
product
Avgas A
Avgas B
Avgas C
Automobile
7.0
7.0
7.0
80.0
91.0
100.0
96.0
130.0
0.5
4.0
4.0
3.0
$4,960
5.846
6.451
4.830
$0.051770
0.409416
0.409416
0.281862
Constituent
Supply
bbl
per day
Constituent Specifications
1cPN
RVP
0.5 cc.
TEL
4.0 cc.
TEL
3cPN
4.0 cc.
TEL
Alkylate
3,800
5.0
94.0
107.5
148.0
Catalytic
2,652
8.0
83.0
93.0
106.0
Straightrun
4,081
4.0
74.0
87.0
80.0
Isopentane
1,300
20.5
95.0
108.0
140.0
Any supplies not used in one of these three aviation gasolines will be used in
premium automobile fuel, the selling price of which likewise appears in the
exhibit. Performance specifications for automobile fuel are not shown since this
product will be composed primarily of constituents not included in this study;
these constituents will be added in the proper proportions to give the desired
performance specifications.
Management has decided to use the entire available supply of the constituents
in one way or another. Their costs can therefore be neglected in selecting the
blending program since they will be the same whatever program is chosen. The
costs of blending itself are also about the same whatever end product is pro
duced and can, therefore, be neglected in solving this problem, too. The only
variable cost factor is the TEL (since some end products use more of this than
others), and its cost per barrel of product is shown in Exhibit VII.
The solution of the problem is given in Exhibit VIII. In the actual case,
however, precise determination of the most profitable blending program
was not the result which was of most interest to the management con
cerned. After all, the company's experienced schedulers could, given suffi
cient time, arrive at programs as profitable or nearly as profitable as those
derived by mathematical programming — although the tests which seemed to
show this were perhaps unduly favorable to the traditional methods because
the schedulers were given the results of the programming calculations in ad
vance, and thus knew what they had to try to attain.
MATHEMATICAL PROGRAMMING INFORMATION FOR DECISION MAKING
77
EXHIBIT VIII.
MOST PROFITABLE PRODUCT MIX
Total
amount
produced
Composed of these constituents:
Product
Alkylate
Catalytic
Straight
run
Isopen
tane
Avgas A
Avgas B
Avgas C
Automobile
5,513
6,207
113
3,800
2,625
27
2,555
1,526
333
854
113
Total
11,833
3,800
2,652
4,081
1,300
The indirect results were what really impressed management. For one
thing, just as in the case of the Heinz Company, it was clear that the time
and effort of experienced personnel would be saved if the job were routinized
by the use of mathematical programming. This, in turn, now made it practi
cal to compute programs for a variety of requirements and assumptions not
previously covered. To illustrate:
The most profitable product mix as shown in Exhibit VIII contains no Avgas
A. However, company policy called for the production of 500 bbl. per day of
this product for goodwill reasons. When the problem was recomputed taking
this factor into account, it was found that the most profitable mix containing
the required 500 bbl. of Avgas A yielded profits about $80,000 per year less
than those resulting from the program of Exhibit VIII.
This loss was considerably higher than management had believed. Presum
ably the cost could have been computed with adequate accuracy by the com
pany's schedulers, but when such calculations are expensive in terms of the time
of senior personnel, they simply do not get made.
"CONCAVE" PROGRAMMING
The field of gasoline refining is perhaps the one in which the most ex
tensive work has been done in trying out actual applications of mathematical
programming to practical operations. One interesting type of «cwlinear pro
gramming has been tried on actual data in this field. The method has been
called "concave" programming.
In our gasoline case, the problem could be solved by linear programming
because it was assumed that the RVP and PN of any product would be a
simple weighted average of the RVP's and PN's of the constituents, the
PN's being calculated for a predetermined amount of TEL in the product.
We have already suggested that under some conditions this assumption is
not strictly true. Linear programming is particularly inapplicable when the
78 PLANNING AND CONTROLLING PRODUCTION LEVELS
problem involves the blending of automotive rather than highgrade aviation
fuels. In such a case it is not at all clear in advance that it will be economical
to use the maximum permitted amount of TEL, and PN is definitely not
proportional to the amount of TEL in the fuel.
The procedures which have been developed to cope with situations like
this have at least approximately solved the problem in a number of actual
cases. 4 The results show the most profitable amount of TEL to use in
various end products as well as the most profitable way to blend the re
finery stocks.
WHAT PROCESSES TO USE
Some of the most perplexing problems of limited resources which man
agement commonly faces do not concern materials but the productive
capacity of the plant. A good example is the problem of choosing what
products to make and what processes to use for manufacturing them when
a shortage of machine capacity restricts production. The problem may
arise because of a shortage of only a few types of machine in a shop which
is otherwise adequately equipped. The SKF Company, for example, has
reported savings of $100,000 a year through the use of scheduling tech
niques developed from linear programming. 5
Rather than describing the SKF application, however, let us take a hypo
thetical example which will give an opportunity to show one of the ways in
which setup costs can be handled by mathematical programming. Setup
costs cannot be handled directly by linear programming because they are not
proportional to volume of production. However, they can be handled indi
rectly by the same means used to deal with the fixed costs that can be
avoided by closing down a plant completely (see the case described under
the heading Where to Produce). Here is an illustrative situation:
A machine shop has adequate machinetool capacity except for three types of
machine, I, II, and III. These machines are used (in conjunction with others) to
make three products, A, B, and C. Each product can be made in a variety of
ways. It is possible, for example, to reduce the amount of time required for
4 See A. S. Manne, Concave Programming for Gasoline Blends, Report P383 of
The Rand Corporation, Santa Monica, 1953.
5 Factory Management and Maintenance, January 1954, pp. 136137. The technique
there described is very close to the "profitpreference procedure" mentioned in the
Appendix, p. 94.
MATHEMATICAL PROGRAMMING INFORMATION FOR DECISION MAKING 79
grinding by closer machining, but this requires more machining time. To be
specific, let us suppose that for each product there are three alternate operation
sheets, which we shall call processes 1, 2, and 3.
If sufficient time were available on all machines, the most economical process
would be chosen for each product individually, and the company would then
make all it could sell of that product. But because of the shortage of capacity
the process to be used for any one product must be chosen with regard to its
effect on machine availability for the other two products, and the quantity to be
produced must be calculated for all products together in such a way as to
obtain the greatest profit from the total production of all products.
The demands of each process for each product on the three critical types of
machine are shown in Exhibit IX; these are preunit times (standards duly
adjusted for efficiency). For example, if Product B is produced by Process 3,
each unit will require 0.2 hour on a machine of Type II and 1.0 hour on a
machine of Type III, but no time on Type I. The weekly available machine
hours are also shown in the exhibit, after deduction of estimated allowances for
repair and maintenance, but with no deduction for setup.
Exhibit IX also shows the number of units of each product which must be
produced each week to fill orders already accepted, together with the "margin"
which will be realized on any additional units that can be produced. This margin
is the selling price less all outofpocket costs of production except the costs of
operating the machines being programmed. Since these machines are the "bottle
necks," they will be used full time or virtually full time in any case, and, there
fore, the costs of operating them will be virtually the same regardless of the
program chosen.
SOLUTION OF PROBLEM
To solve the problem, we start by neglecting the setup times for the ma
chines (shown in Exhibit IX) just as we first neglected fixed costs in de
ciding where to produce ketchup. We simply deduce a roughly estimated
flat six hours from each of the weekly machine availabilities and then de
velop a program based on the assumption that any program would involve
exactly six hours total setup time on each type of machine. We can subse
80
PLANNING AND CONTROLLING PRODUCTION LEVELS
quently adjust for the number and kind of setups actually called for by the
program.
EXHIBIT IX. MACHINESHOP REQUIREMENTS
Machine type
A — PerUnit Machine Times
I
II
III
Product
A
A
A
B
B
B
C
C
C
Process
1
2
3
1
2
3
1
2
3
Machine hours per unit
0.2
0.2
0.2
0.4
—
0.3
0.6
0.1
0.1
0.2
0.3
0.4
0.1
0.1
0.8
—
0.2
1.0
0.2
0.1
0.7
0.1
0.6
0.4
—
0.8
0.2
B — Total Machine Hours Available per Week
Machine type I II III
Hours
Product
118
230
306
Product Requirements and "Margins"
A B
Minimum units
per week
Margin per uni
production
required
t on additional
100
$10
200
$20
300
$30
Machine type
D — Machine
Setup Times
I
//
///
Product
Process
Machine
hours
per setup
A
A
A
1
2
3
2.4
1.8
1.2
0.6
1.8
1.2
1.8
1.2
B
B
B
1
2
3
3.0
0.6
1.2
3.0
3.6
2.4
1.2
1.2
C
C
C
1
2
3
2.4
1.2
1.8
1.2
2.4
3.0
1.2
2.4
Exhibit X shows the program which would be the most profitable if this
assumption concerning setup were true. It calls for the production of only
the required 100 units per week of Product A and 200 units of B, but it calls
for 394 units of Product C instead of just the required 300. In other words,
the calculation indicates that the most profitable use which can be made of
the available capacity after fulfilling contractual obligations is to produce
Product C.
Checking to see how much setup time is actually implied by this program,
we discover that it exceeds the sixhour estimate on all three types of
MATHEMATICAL PROGRAMMING INFORMATION FOR DECISION MAKING 81
machine (see the totals shown in the table under B, Exhibit X). We could
adjust for this by simply reducing the available machine hours accordingly
and then recalculating the program, but examination of the program of
Exhibit X brings to light another fact of which we ought also to take ac
count. This is the fact that only 8 units per week of Product A are to be
manufactured by Process 3 .
EXHIBIT X. MOST PROFITABLE USE OF CAPACITY ASSUMING
SIX HOURS SETUP PER MACHINE
A — Program Based on Six Hours Setup per Machine
Machine
type
/
II
III
Units
Product
Process
1
Productive machine hours
produced
A
18.4
18.4
18.4
92
A
3
4.8
0.8
0.8
8
B
1
40.0
60.0
80.0
200
C
1
48.8
24.4
170.8
244
C
3
—
120.0
30.0
150
Total
112.0
223.6*
300.0
B — Actual Setup Times Implied by Program
Machine
type
/
//
///
Product
Process
1
Hours of setup time
A
2.4
0.6
1.2
A
3
1.2
1.8
1.2
B
1
3.0
1.2
2.4
C
1
2.4
1.8
3.0
C
3
—
2.4
2.4
Total
9.0
7.8
10.2
* Discrepancy from 306.0 due to rounding of figures. .
Since these are bottleneck machines, we do not really need a cost calcula
tion to decide that it is wasteful to tie them up in setup for this almost
negligible amount of production. (This decision can be checked, as will be
shown shortly.) Therefore we eliminate Process 3 for Product A before
adjusting the available machine hours for the amount of setup time actually
required, and then recalculate the program, again excluding the unwanted
process. One of the more useful features of linear programming is the fact
that the calculation need not be purely mechanical, but can always be con
trolled to agree with common sense.
The resulting revised program is shown in Exhibit XI, together with
some related cost information which corresponds to the "row values" and
"column values" of the ketchup problems. This information will be dis
cussed more fully in Part III of this article. For the moment we may observe
that it confirms our decision to reject Process 3 for Product A. Use of this
process for 8 units would save running time worth $51.20 (8 X $6.40) but
82
PLANNING AND CONTROLLING PRODUCTION LEVELS
would cost nearly $100 in setup (1.8 hours on a Type II machine worth
$27.80 per hour plus 1.2 hours on a Type III machine worth $38.80 per
hour).
We could at this point ask whether it might also be better to use only a
single process for Product C. Common sense tells us, however, that the pro
duction of Product C by each of the two methods is large enough to make
setup cost negligible, and again this can be confirmed by analysis of the
byproduct cost information and other data on the worksheets underlying
EXHIBIT XI. MOST PROFITABLE USE OF AVAILABLE CAPACITY
A — Revised Program Based on Actual Setup Requirements
Machine
type
I
II
Ill
Product
Process
1
setup
Machine hours
Units
produced
A
2.4
0.6
1.2
100
run
20.0
20.0
20.0
B
1
setup
3.0
1.2
2.4
200
run
40.0
60.0
80.0
C
1
setup
2.4
1.8
3.0
238
run
47.6
23.8
166.6
C
3
setup
—
2.4
2.4
150
run
—
120.2
30.0
Idle time
2.6
—
—
Total
118.0
230.0
305.6 =
B — Additional Margin Which Would Be Made Possible
by One Additional Machine Hour
Machine type I II III
Margin
Product
A
B
C
Product
A
B
$27.80
$38.80
C — Loss of Margin Which Would Result from Production
of One Unit by Processes Other than Those Selectedf
Process
2
$(1.70)$
10.00
2.20
3
$(6.40)$
20.80
D — Loss of Margin Which Would Result from Production
of One Extra Unit of Product Other than Product C
Loss
$3.30
3.00
* Discrepancy from 306.0 due to rounding of figures.
+ This table gives the loss which would arise from the running time of the process in question.
The loss due to setting up for the additional process can be calculated from the value of one ma
chine hour shown in the previous table.
t Minus quantity.
Exhibit XI. However, the argument is a little more complex than the one
concerning Process 3 for Product A and will not be given here.
MATHEMATICAL PROGRAMMING INFORMATION FOR DECISION MAKING 83
FEATURES OF PROGRAM
The final program still calls for only the required amounts of Products A
and B; proper choice of processes for all products makes it possible to pro
duce 88 units per week of Product C above the minimum requirements.
This figure of 88 units is not greatly different from the 94 units shown in
the firstapproximation program (Exhibit X). That program, despite the
roughandready assumption on which it was based, proved in fact to be a
very good guide to the proper use of the available capacity, and only minor
refinements were required to make it into the genuinely most profitable
program. A more complex problem might, of course, call for several suc
cessive approximations instead of just two as in this simple case.
One significant feature of the final program is the fact that it calls for a
certain amount of idle time on machines of Type I. Any program which
used this type of machine fully would produce less profit than the program
of Exhibit XI. In one actual application of mathematical programming to a
machine shop, a result of exactly this sort proved to be of very considerable
practical importance. Without some kind of provable justification, personnel
were extremely hesitant to include idle time in the program when manage
ment was pressing for all possible production. There is a real danger under
such conditions that personnel will produce a program less efficient than is
possible simply because they concentrate their efforts on discovering a pro
gram which uses all machines 100% of the time.
LOWEST COST PRODUCTION
The last few examples have involved the problem of getting out the most
profitable production when a company can produce less than it can sell.
Mathematical programming can also be of value when the problem is one of
getting out the required production at the lowest possible cost. Here is an
interesting example:
One of the large meat packers is currently using linear programming to find
the least expensive way of producing a poultry feed with all the required nutri
tive values. All that is needed to solve such a problem is: a list of the essential
nutrients (minerals, proteins, and so forth) with the amount of each which
should be contained in a pound of feed; a list of the possible materials which
could be used to produce the feed, with the price of each; and a table showing
the amount of each nutrient contained in a pound of each possible constituent
for the feed. 6
This problem is obviously very similar to the Avgas problem discussed
above, except that here the object of the program is to supply a fixed output
G The use of mathematical programming in connection with a variety of problems in
farm economics is described in a number of articles in the Journal of Farm Eco
nomics, 1951, p. 299; 1953, pp. 471 and 823; 1954, p. 78.
84 PLANNING AND CONTROLLING PRODUCTION LEVELS
at lowest cost rather than to choose the output which will Liaximize revenue.
Exactly the same kind of problem can arise when there is more than a
single end product involved. For instance, the manager of a refinery might
be faced with this kind of problem:
Suppose that instead of having inadequate supplies, this manager has ample
capacity to make all he can sell. As we have seen, each of the products which
he sells can be blended in a variety of ways from intermediate products such as
alkylates and catalyticcracked gasolines, and each of these intermediate prod
ucts can be produced out of various crudes in various proportions. The manager
of the refinery must decide which crudes to buy and how they should be refined
so as to produce the required end products at the lowest possible cost.
Charnes, Cooper, and Mellon have shown that it is possible to use linear
programming to solve a still more complex problem than this, bringing in,
for example, the possibility of using imported as well as domestic crudes,
and considering even such factors as taxes, customs duties, and the cost
differences between chartered and companyowned tankers. 7
Programming can also assist in cost reduction in a machine shop when
there is sufficient capacity to produce all that can be sold of every product;
it can indicate how to produce each product by the most economical
process. All that is required for a programming problem to exist is that the
capacity of the company's best or most economical machines of a given
type — for example, its highestspeed screw machine — be less than sufficient
for the entire production requirements. To illustrate:
Suppose that a manufacturer wants to produce specified quantities of five
different screwmachine parts, A through E, and has available three different
EXHIBIT XII.
PRODUCTION RATES, REQUIREMENTS,
AND COSTS
Machine
/ // ///
Average weekly
production
(units)
Part
Perunit machine time
(minutes)
A
B
C
D
E
0.2 0.4 0.5
0.1 0.3 0.5
0.2 0.2 0.4
0.1 0.3 0.3
0.2 0.3 0.5
Variable operating cost
(Per hour)
4,000
9,000
7,000
9,000
4,000
$12 $9 $9
screw machines, I, II, and III. Anv of the machines can produce any of the
parts, but the rates of operation are different, as shown by the perunit times in
7 A. Charnes, W. W. Cooper, and B. Mellon, "A Model for Programming and
Sensitivity Analysis in an Integrated Oil Company," circulated in mimeographed form
by the Carnegie Institute of Technology, and to be printed in a forthcoming issue of
Econometrica.
MATHEMATICAL PROGRAMMING INFORMATION FOR DECISION MAKING 85
Exhibit XII. If Machine II were slower than Machine I by the same percentage
on all parts, and the same were true of Machine III, this problem would not
require much thought for its solution, but when the inferiority of a machine
depends on the particular part, linear programming is of use.
The hourly variable cost (direct labor, power, repair and maintenance, etc.)
of operating each machine is shown in the exhibit, since the machines are not
all bottlenecks and the whole point of the problem is to avoid operating costs
insofar as possible. The exhibit also gives the required average production of
each part on a weekly basis, though we shall assume that management can make
each part in long runs and thereby reduce setup cost to a point where it may
be neglected in determining the program. Setup, maintenance, and repairs we
shall assume to be performed on Saturday, and therefore we take each machine
as being available 40 hours per week.
The lowestcost program which will accomplish the required production is
shown in Exhibit XIII together with the usual byproduct cost information. As
EXHIBIT XIII. LOWESTCOST PROGRAM AND BYPRODUCT
COST INFORMATION
A — LowestCost Machine Assignments
First Alternat
ive
Sec
ond A Iternative
Program
Program
Machine
/
II
III
I
II
///
Part
A verage
weekly
minutes
A
600
500
467
833
B
900
900
C
1,400
1,400
D
900
900
E
1,000
333
133
1,000
Idle time
1,567
1,567
Total 2,400 2,400 2,400 2,400 2,400 2,400
B — Cost of One Additional Unit of Product
A
BCD
E
Cost $0.0750
Machine
$0.0375 $0.0500 $0.0375
C — Value of One Additional Machine Hour
I II
$0.0750
III
Value
$10.50 $7.50
$0.00
previously stated, the production shown in the exhibit is in terms of weekly
averages; the actual length of individual runs can be determined subsequently,
in the usual way in which economic lot sizes are determined.
PART III. COST AND PROFIT INFORMATION
Determination of the most profitable program under a particular set of
circumstances is by no means the only advantage which management can
derive from the intelligent application of mathematical programming. In
86 PLANNING AND CONTROLLING PRODUCTION LEVELS
many situations the technique will be of equal or even greater value as the
only practical way of obtaining certain cost and profit information that is
essential for sound decisions on both shortrun and longrun problems of
many kinds.
NEED FOR PROGRAMMING
What kind of cost information will mathematical programming provide?
The gasoline blending case described in Part II of this article is a good
example.
In that instance the management learned that the manufacture of Avgas
A was leading to a reduction of nearly $80,000 a year in profits, far more
than had been believed. Now, "cost" in this sense — the difference between
the profit which results from one course of action and the profit which
would result from another course of action — is obviously a completely
different thing from cost in the accounting sense. Information regarding this
kind of cost cannot be provided by ordinary accounting procedures. In fact,
mathematical programming is the only way to get it quickly and accurately
when there are many possible combinations of the various factors involved.
COSTS FOR DECISION MAKING
In some situations the need for looking at the effect of a proposed action
on overall profits rather than at its accounting cost or profit is perfectly
clear. In our gasoline blending case, management knew very well that
money was being lost by the production of Avgas A even though the
accounts showed a profit; it was only the extent of the loss that was un
known. In other situations, by contrast, accounting cost is really misleading
in arriving at a sound decision, and it is easy to overlook this fact. An
example should help make this point clear:
It would seem to be plain common sense that the cost of freight to a particular
warehouse is simply the freight bill which is paid on shipments to that ware
house. But management will do well to think twice before acting on the basis
of this "commonsense" view.
Suppose that the sales manager of the company whose shipping program is
given earlier in Exhibit II finds that it is becoming very difficult and expensive to
sell the supply allocated to Warehouse E, whereas sales could easily be increased
at Warehouse T. Selling price is the same at both localities and, because of
competition, cannot readily be changed. On inquiry the sales manager finds that
Warehouse E is being supplied at a freight cost of 23 cents per cwt., whereas
freight to Warehouse T is only 6 cents per cwt. He proposes, therefore, that
supplies and sales be diverted from E to T, thus increasing the company's profits
by the freight saving of 17 cents per cwt. as well as reducing the cost of
advertising and other selling expense.
MATHEMATICAL PROGRAMMING INFORMATION FOR DECISION MAKING 87
The traffic manager will probably counter that the two warehouses are not
being supplied from the same factory, and that if the supplies now being sent
from Factory II to Warehouse E are shipped to Warehouse T instead, freight
costs will not fall to 6 cents per cwt., but will increase from the present 23 cents
to 54 cents, making a loss of 31 cents per cwt.
Actually, neither of the two would be right. In the event that supplies are
diverted from Warehouse E to Warehouse T, there will in fact be an extra
freight cost rather than a saving. But if the change is properly programmed (the
supplies formerly sent from II to E should be sent to Q, which can then take
less from XII, which in turn can then supply the additional amount to T), then
the extra cost will be only 14 cents per cwt. It is this cost which management
should compare with the estimated extra cost of selling at Warehouse E.
The example just cited and the gasoline blending case are typical of the
way in which mathematical programming can be used to calculate the cost
or profit which results or will result from a management decision. Generally
speaking, any program is determined in such a way as to produce the
greatest possible profits under a certain set of fixed conditions. If manage
ment wishes to consider a change in any of these conditions, a new program
can be computed and profits under the two sets of conditions can then be
compared.
AVAILABLE FIGURES
In some cases it is not even necessary to compute a new program to find
the cost or profit which applies to a proposed decision. The computation of
the original program itself yields as a free byproduct the cost or profit
which will result from certain changes in the conditions underlying the
program, provided that these changes are not too great in extent. In the
jargon of the economists, these byproduct figures are "marginal" cost or
profit rates. To illustrate:
For diversion of sales from Warehouse E to Warehouse T, the marginal cost
is given immediately by comparison of the "row values" shown in Exhibit II for
the two warehouses. The value for E is 28 cents per cwt., the value for T is 42
cents, and the extra cost is therefore 14 cents per cwt. (4228). We can be sure
at once that this will be the extra cost if only a single cwt. is diverted from one
warehouse to the other, but in order to find the cost of a larger diversion we
must study the program itself. If we do so, we will find that the marginal rate
will hold in this case even if the entire supply now allocated to E is diverted to
T. If, on the contrary, we were considering diversion from Warehouse G to T,
we would find that the marginal rate of 15 cents (4227) would apply only to
the first 1 80 cwt.
The "column values" of Exhibit II give similar information concerning the
cost or saving which will result from shifting production from one plant to
another. If production is increased at Factory V and decreased at Factory VI,
88 PLANNING AND CONTROLLING PRODUCTION LEVELS
there will be a saving of 13 cents per cwt. (—38— [—51]) up to a certain limit,
and study of the program shows that this limit is again 180 cwt.
The costs shown in Exhibit XI and XIII are marginal rates of this same
sort. In fact, such information could have been given in connection with all
the programs developed in this article.
Probably the most important use of the marginal rates is that they im
mediately give a minimum figure for the cost of a change which reduces
profits, or a maximum figure for the profitability of a change which increases
profits. For example, when the program of Exhibit XI shows that an addi
tional hour on a machine of Type III is worth $38.80, we can be sure that
ten additional hours will be worth no more than $388, although they may
be worth less. Inspection of the marginal costs can thus be of practical
value in limiting the range of alternatives which are worth further investiga
tion.
USES OF INFORMATION
Now let us turn to consider a number of examples of particular kinds of
cost and profit information which can be obtained by mathematical pro
gramming and which will be of use in making management decisions.
PRODUCT COST
The gasoline blending case was as good an illustration as possible of the
use of mathematical programming to find the true profitability of a particular
product, but the technology of gasoline blending is so complex that it is not
easy to see why the answer comes out as it does. Since it is difficult to make
intelligent use of a technique without really understanding how it operates,
let us look briefly at a much simpler example of the same kind of problem:
In the first case involving the assignment of machine tools in Part II, there
was idle capacity available after meeting the contractual commitments (see
Exhibit XIII). Suppose that, after this schedule has been worked out, a customer
places an order for an additional 1,000 units of screwmachine Part D. What
will be the cost of filling this order?
Machine III is the only machine with idle capacity; and if the additional
quantity of Part D is made on that machine, it will cost $75 (500 minutes at $9
per hour). The most economical course of action, however, is to produce the
additional 1,000 units of D on Machine I, obtaining the required 100 minutes
by taking 500 units of Part A off this machine and putting them on Machine III.
If this is done, the accounting cost of the 1,000 units of D will be only $20
(100 minutes at $12 per hour), but the actual addition to total cost will be
$37.50 (250 minutes at $9 per hour to make the 500 units of A on machine
III). Thus the true cost of the additional 1,000 units of D will be $0.0375 each,
the value shown in Exhibit XIII. Any price above the sum of this figure and the
material cost of the part will make a contribution to fixed overhead.
MATHEMATICAL PROGRAMMING! INFORMATION FOR DECISION MAKING 89
MOST PROFITABLE CUSTOMERS
The example of the diversion of sales from Warehouse E to Warehouse T
previously discussed shows how programming can be used to determine
which customers are the most profitable in a situation where the only differ
ence among customers lies in the cost of freight. The question would be no
harder to answer if some customers were supplied from plants with higher
production costs than others. Actually, of course, there is very little differ
ence between determining the profitability of a product and the profitability
of a customer.
MARKETING POLICY
Cost and profit information calculated by mathematical programming can
be of use to management in deciding what products to make, what prices to
set, and where to expend selling effort. We wish to emphasize, however,
that we are not proposing that management should build its entire marketing
program on the basis of shortrun profit considerations. Programming pro
vides information; it does not provide answers to policy questions.
On learning that certain products or certain customers are relatively un
profitable under present conditions, it is up to management first of all to
decide whether the situation is temporary or likely to continue for some
time to come. This means that management should forecast future costs
and future sales potentials under a variety of reasonable assumptions, and
then calculate the profitability of the various products or markets under
various combinations of these assumptions. It is here that mathematical
programming will make its real contribution, since it is only when such
calculations can be easily and cheaply carried out that management can
afford to investigate a wide range of assumptions.
After such calculations have been made, management can decide to
change prices, refuse certain orders, accept them at a shortrun loss, or
install new capacity of such a kind and at such places that the products or
markets in question will become profitable.
COST OF IMPROVEMENTS
Another kind of cost which it is often important to know is the cost of
an improvement in the quality of product or service rendered to the cus
tomer. A similar problem arises when it is necessary to decide whether im
proved materials acquired at higher cost will increase revenues or reduce
other costs sufficiently to justify their higher cost. Here are some illustrative
cases:
1 . Cost of quick delivery — According to the shipping program of Exhibit
II, Warehouse M is to be supplied partly from Factory II at a cost of 40
90 PLANNING AND CONTROLLING PRODUCTION LEVELS
cents per cwt. and partly from Factory IV at 21 cents per cwt. Suppose
that stocks are low at this warehouse and that the manager would like to
obtain some supplies quickly from the nearest source, Factory V. Since this
is the nearest plant, the freight rate to Warehouse M, 10 cents per cwt., is
naturally lower than the rates from the factories currently supplying the
warehouse; but use of this shorter route will necessarily result in an increase
in total cost, since the program as it stands gives the lowest possible total
cost.
Programming shows immediately that the extra cost will be 16 cents per
cwt. for the first 140 cwt. shipped to M from Factory V. The higher cost
applying to additional quantities could be readily calculated if it were
needed.
2. Choice of process in a machine shop — In the case of the machine
shop with limited total capacity, Exhibit XI showed that the most profitable
course of action was to produce Product B by the use of Process 1 . Suppose
that while an adequate product results from this process, a better quality
would result from the use of Process 3. Would it be worth using this process
in order to increase customer satisfaction, or could the price be increased
sufficiently to recover a part of the additional cost?
The program of Exhibit XI shows immediately that the extra cost result
ing from the use of Process 3 for Product B will be at least $20.80 per unit.
The cost arises because use of this process instead of Process 1 takes up
capacity which is being used for the production of Product C, each unit of
which produces a "margin" of $30 per unit. Up to 128 units of B can be
made by Process 3 instead of Process 1 at the cost of $20.80 per unit. If
128 units are made, the entire capacity of the shop will be used up in pro
ducing the contractual commitments for the three products, and further use
of Process 3 for Product B will be impossible.
3. Cost of antiknock rating — In the gasoline refinery studied by Charnes,
Cooper, and Mellon, antiknock ratings (PN's) were specified for Avgas B
and Avgas C for both rich and lean mixture. During the study an interesting
question was raised as to the additional cost entailed by the richmixture
specification. It was found to amount to over $1,000 per day. In other
words, profits could have been increased by that amount if only a lean
mixture rating had been required in the products. A little further calculation
with their data produced the equally interesting result that the leanmixture
requirement on these two fuels was costing nothing; satisfaction of the rich
mixture requirement automatically produced oversatisfaction of the lean
mixture requirement.
4. Value of improved materials — Engineers of this same refinery sug
gested that if the volatility of the straightrun gasoline being used in blend
ing could be reduced, it would be possible to produce a product mix with a
MATHEMATICAL PROGRAMMING INFORMATION FOR DECISION MAKING 91
considerably higher market value. Again, programming provided significant
and accurate information. It was able to show that if the RVP of this stock
could be reduced by one unit, from 4.0 to 3.0, the market value of the
products could be increased by $84 per day. Thus, if the improved stock
could be produced at an additional cost smaller than this, it would pay to
do so; otherwise it would not.
CAPITAL INVESTMENTS
Some of the most important decisions that management has to make are
those which involve the choice of the most profitable ways in which to invest
new capital. The choice is usually made by comparing the cost of each
proposed investment with the increase of income that it will produce. When
several of the proposed investments are for use in the same productive
process, and when this process produces a variety of different products, it
may be extremely difficult to determine the additional income that will
result from any one investment or from any combination of investments
without the use of a systematic computing technique.
Machine Tools. Consider, for example, the machineshop case described
in Part II in which sales were limited by machine capacity. Under the pro
gram of Exhibit XI, all machines of Type II and Type III are loaded to
capacity; and while there is idle time on machines of Type I, it is very small
in amount and actually exists only because it was unprofitable to set up to
produce just 8 units per week of Product A by Process 3. Under these con
ditions what would be the return on an investment in an additional machine
of one of the three types? It will be enough to work out the answer to this
question for just one of the three types as an example, assuming that man
agement has forecast that present demand and present costs and prices will
remain unchanged in the future:
Suppose that if the shop acquires one additional machine of Type III, it
would be available for 38 hours per week (one shift with allowance for down
time). We simply calculate a new program for the same conditions as shown
in Exhibit IX, except that we increase the available time on machines of Type
III from 300 to 338 hours. The resulting program shows a $960 perweek in
crease in "margin" — selling price less all costs of production except the costs on
the bottleneck machines. (To find the additional income produced by the new
machine, we would have to subtract the labor and overhead costs of operating
the machine and the depreciation and other costs of owning it.)
The result is due to the fact that the additional machine will make it possible
to produce 32 additional units of Product C per week. Note that the $960
margin on 38 hours of use amounts to only $25.30 per hour, considerably less
than the $38.80 shown in Exhibit XI. As more time is made available on
machines of Type III, the bottleneck on this type becomes relatively less im
92 PLANNING AND CONTROLLING PRODUCTION LEVELS
portant and the bottlenecks on the other two types become relatively more
important.
Raw Materials. Without actually working out examples, we can point to
either the gasoline refinery or the hypothetical case on the selective use of
raw materials (both in Part II) as two other situations where the profit
ability of investment would be very difficult to calculate without the use of
mathematical programming. The refinery problem discussed above involved
only the most profitable way in which to blend existing supplies of materials.
Mathematical programming would readily show the additional sales revenue
which could be obtained (at present prices) if the refinery were to enlarge
its facilities for production of one or more of the blending stocks.
In the case on selective use of raw materials, the materials had to be
purchased in the market; and, as shown in Exhibit VI, it proved unprofit
able to produce Product D because of the limited supplies of materials
available at normal prices. Programming could readily show how much the
company could afford to invest in a source of raw materials in order to
obtain them at more reasonable cost.
Programming and Forecasts. In the case of investment decisions even
more than in the case of the other types of decisions previously discussed,
the relevant data are not so much the facts of the immediate present as they
are forecasts of conditions which will prevail in the future. An investment
decision cannot be made rationally unless it is possible to explore its profit
ability under a variety of assumptions about future costs and markets.
It is already difficult enough to make the necessary forecasts; without the
use of a systematic technique for calculation, full exploration of their impli
cations is virtually impossible because of time, trouble, and expense. It is
for this reason that it seems likely that mathematical programming may be
of even greater value to management in the field of planning than in the field
of immediate operating decisions.
As in the case of its other applications, however, mathematical program
ming is not a cureall. Management can use it to great advantage in planning
and policy making, but executives must first understand it correctly and be
able to use it intelligently in combination with the other tools of forecasting
and planning. The fate of mathematical programming, in other words, lies
today in management's hands. The scientists, the inventors, have done their
work; it is now up to the users.
Appendix
DIRECTIONS FOR SOLVING PROBLEMS
BY A USEFUL SHORT PROCEDURE
There are several alternate procedures available for solving problems of linear
programming. One of these will work in all cases but takes a long time to carry out —
the "general procedure," which is discussed toward the end of this appendix. The
others are relatively quick, but will work only in certain cases — e.g., the "profit
preference procedure" and the "transportationproblem procedure."
MATHEMATICAL PROGRAMMING INFORMATION FOR DECISION MAKING 93
A very restricted class of problems can be solved by hand with remarkable ease
through the use of the "profitpreference procedure." A good example of its use is the
scheduling of two classes of machine tools which formed a bottleneck in the opera
tions of one actual company. The example has been published, with clear instructions
for carrying out the procedure. 8
By far the most frequently useful of the shorter procedures is the one known as the
"transportationproblem procedure." 9 As pointed out in the preceding text, it got this
name because it was developed to determine lowestcost shipping programs, but it can
be used for problems not involving transportation (just as certain problems involving
transportation cannot be solved by it). Because of its simplicity, we shall give full
directions for its use, first working through a simple example and then giving some
suggestions for reducing more complex problems to such a form that they can be
solved in the same way.
TransportationProblem Procedure
Our example consists of assigning the production of three plants to fill the require
ments of four warehouses in such a way that the total cost of freight will be at a
minimum. This example involves so few variables that it could be solved far more
quickly by common sense than by the use of a formal procedure. The example is
adequate, nevertheless, to explain the procedure, and the procedure can then be used
to solve much larger problems that would be extremely difficult to solve by common
sense. Furthermore, the procedure itself can be considerably shortcut once it is
understood; some suggestions for doing that will be given.
Table A gives the data for the problem: the freight rates from each plant to each
warehouse, the capacity of each plant, and the requirements of each warehouse. Now
let us go through the various steps of the solution.
TABLE
A.
RATES,
REQUIREMENTS,
AND CAPACITIES
Factory
/
//
///
Warehouse re
quirements (tons)
ns)
Freight
rates {dollars
per
ton)
Warehouse A
B
C
D
Factory capacity (tc
1.05
2.30
1.80
1.00
5
.90
1.40
1.00
1.75
60
2.00
1.40
1.20
1.10
40
35
10
35
25
105
Getting a Starting Program. We first get a shipping program which satisfies the
fixed requirements and capacities, regardless of cost, by the following procedure. Take
Factory I and assign its 5 tons of capacity to Warehouse A. Fill the remaining 30 tons
of this warehouse's requirements from Factory II. Then use 10 more tons of Factory
II's capacity to satisfy Warehouse B, and assign its remaining 20 tons in partial
satisfaction of Warehouse C. Complete C's requirements from Factory III, and use
the remainder of Ill's capacity to satisfy Warehouse D. This produces the starting
program of Table B. The procedure could obviously be used to assign warehouses
to factories in a problem of any size.
A starting program can be based on a guess at the best solution rather than on
the "blind" procedure described in the text; and if the guess is any good at all, subse
quent calculation will be materially reduced. Start with any factory at all and use its
8 See A. Charnes, W. W. Cooper, and D. Farr, "Linear Programming and Profit Prefer
ence Scheduling for a Manufacturing Firm," Journal of the Operations Research Society of
America I, May 1953, pp. 114129. (The reader should be warned that errors have crept into
Tables HI and IV of this publication.) The technique is similar to the one used by SKF:
cf . above, p. 86.
9 This procedure was developed by G. B. Dantzig: see T. C. Koopman's, Activity Analysis
of Production and Allocation (New York, John Wiley & Sons, Inc., 1951), pp. 359373.
94
PLANNING AND CONTROLLING PRODUCTION LEVELS
TABLE B.
INITIAL PROGRAM
(Tons)
OF SHIPMENTS
Factory
II
III
Warehouse A
B
C
D
Total
30
10
20
60
15
25
40
Total
35
10
35
25
105
capacity to fill the requirements of those warehouses which it seems most economical
to assign to this factory. When that factory's capacity has been used up, take any
other factory; first use its capacity to complete the requirements of the warehouse
which was left only partially satisfied at the end of the previous step, and then go on
to fill any other warehouses which it seems sensible to assign to the second factory.
The only rule which should not be neglected is to finish filling the requirements of
one warehouse before going on to a new one. If the number of plants is greater than
the number of warehouses, it is perfectly legitimate, however, to reverse the procedure.
Start by assigning one warehouse to a series of plants, and, when the warehouse's re
quirements are filled, take the next warehouse, use it to absorb the leftover capacity
of the last factory previously used, and then go on to new factories.
The easiest way to do the work is on paper ruled into squares; and in the following
discussion reference is made to locations in the tables as "squares"; for example, the
number located in Row B and Column III is said to be in Square B III.
Row Values and Column Values. Next build up a "cost table" by the following
procedure:
(1) Fill in the actual freight rates, taken from Table A, for those routes which
are actually in use in Table B. This produces Table C except for the "row values"
and "column values."
(2) Fill in the "row values" and "column values" shown in Table C. To do this,
assign an arbitrary row value to Row A; we have chosen .00 for this value, but it
might have been anything. Now under every square of Row A which contains a rate,
assign a column value (positive or negative) such that the sum of the row and
column values equals the value in the table. In Column I we put a column value of
1.05, since 1.05 + .00 gives the value 1.05 found in Square A I; in Column II we put
a value of .90, since .90 + .00 gives the .90 in Square A II.
(3) We have now assigned all the column values which we can assign on the basis
of the row value for Row A. We must next assign additional row values on the basis
of these column values. We therefore look for rows with no row value but containing
rates in squares for which column values exist. We observe that Rows B and C both
have rates in Column II, which has a column value of .90. The row value for Row B
must be set at .50, since .90 + .50= 1.40, which is the rate in B II. By the same
reasoning, we arrive at .10 as the row value for Row C.
TABLE C.
RATES FOR ROUTES USED IN TABLE B
(Dollars per ton)
Factory
II
III
Warehouse A
B
C
D
Column value
1.05
1.05
.90
1.40
1.00
.90
1.20
1.10
1.10
Row value
.00
.50
.10
.00
(4) No further row values can be assigned, so we go back to assigning column
values by looking for rates which now have a row value but no column value. We
MATHEMATICAL PROGRAMMING INFORMATION FOR DECISION MAKING 95
observe that there is a 1.20 in Square C III, which has a row value of .10 but no
column value. The column value must be 1.10 in order to have 1.10 + .10= 1.20.
(5) Finally, we assign the one missing row value. In Row D there is 1.10 in
Square D III, with a column value of 1.10 and no row value. The row value must be
.00 if the total of the row and column values is to equal the value in the square.
This procedure of alternately assigning row and column values can always be ex
tended to fill in the row and column values for any cost table provided that "de
generacy" is not present in the corresponding route table. Degeneracy will be ex
plained and a method of dealing with it will be described subsequently. In the absence
of degeneracy, inability to complete the row and column values, or the existence of
contradictory evidence on row and column values, indicates that an error has been
made either in drawing up the table of routes (Table B) or in putting down in the
cost table (Table C) the rates which correspond to the routes in Table B. On the
other hand, it is not essential to derive the row values in the order A, B, C, D and
the column values in the order I, II, III; they may be derived in any order that is
possible.
The Cost Table. We now proceed to make Table C into a complete cost table,
Table D, by filling in all the blank squares with the total of the appropriate row and
column values. For example, the 1.55 in Square B I is the total of the row value for
Row B (.50) and the column value for Column I (1.05). The figures thus derived are
shown in Table D in lightface type, whereas the figures taken from Table C and
corresponding to routes actually in use (in Table B) are shown in boldface type. (In
practice, the cost table can be made up directly without actually filling in the row and
column values.)
TABLE
D.
COSTS
FOR ROUTES
(Dollars per ton)
USED
IN TABLE
B
Factory
/
II
III
Ro
w value
Warehouse A
B
C
D
Column value
1.05
1.55
1.15
1.05
1.05
.90
1.40
1.00
.90
.90
1.10
1.60
1.20
1.10
1.10
.00
.50
.10
.00
Revising the Program. We now have a complete set of tables: a rate table, a route
table, and a "cost" table. We proceed to look for the best change to make in the route
table in order to reduce the cost of freight. To find this change, we compare the cost
table, Table D, with the rate table, Table A, looking for the square where the figure
in Table D is larger than the corresponding figure in Table A by the greatest differ
ence. This is Square B III. The fact that Table D shows 1.60 while Table A shows
1.40 tells us (for reasons to be explained later) that if we make shipments from
Factory III to Warehouse B, and make the proper adjustments in the rest of our
program, we shall save 20 cents for every ton we can ship along this new route.
The next problem is to find out what adjustments will have to be made in the rest
of the program and, thereby, to find out how much we can ship along the new route
from III to B. To do this, we construct Table E by first copying Table B (in actual
practice there would be no need to copy the table) and then going through the
following procedure.
(1) In the Square B III write +x: this is the as yet unknown amount which will
be shipped over the new route from III to B. We have now overloaded the capacity
of Factory III by the amount x, and must therefore decrease by x the amount which
III is to supply to some other warehouse. When this is done, it will be necessary to
supply this warehouse from some other factory, and so on.
96 PLANNING AND CONTROLLING PRODUCTION LEVELS
(2) To locate the factories and warehouses which will not be affected, look
through Table E and put a star beside any number which is the only number in
either its row or its column, but remember that the x in B III counts as a number.
This leads to putting a star beside the 5 in A I and the 25 in D III. Considering the
starred numbers as nonexistent, look through the table again and put a star beside
any numbers which are now left alone in their row or column owing to the elimina
tion of the starred numbers in the previous step. This leads to putting a star beside
the 30 in A II, since with the 5 in A I starred, A II is alone in its row.
Now look through the table again for additional numbers which have been left
alone in their row or column. In this case we can find none, so the operation is com
plete; otherwise, we would continue eliminating until no more isolated numbers
could be found.
TABLE E. CHANGES TO BE MADE IN ROUTES OF TABLE B
(Tons)
Factory I H /// Total
Warehouse A 5* 30* 35
B 10x +x 10
C 20+x 15x 35
D 25* 25
Total 5 60 40 105
(3) Having completed the foregoing procedure, we now make all required adjust
ments by changing the amount to be shipped along those routes which have not been
eliminated by a star. (Once a little experience has been gained, the routes affected by
a change can easily be found without first starring the routes not affected.) The +x
in B III overloads Factory III, so write — x beside the 15 in C III. Warehouse C is now
short by x, so write +x beside the 20 in C II. Factory II is now overloaded, so write
—x beside the 10 in B II. This last — x balances the fx in Row B with which we
started, so that the effect of using the new route has been completely adjusted for
throughout the program.
(4) Since we shall save 20 cents for every ton we ship along the new route from
III to B, we wish to divert as much tonnage as possible to this route. We therefore
look at all the squares in which we have written — x and discover that the smallest
number with — x beside it is the 10 in B II. This is the limit to the diversion, and
therefore the value for the unknown x. We now produce Table F by subtracting 10
in Table E wherever — x was written and adding 10 wherever + x wa s written. This
is our first revised program of shipments. By multiplying the shipments along each
TABLE F. FIRST REVISED PROGRAM OF SHIPMENTS
(Tons)
Factory I II III j ota \
Warehouse A 5
B
C
D
Total 5 60 40 105
route by the rate for that route, the reader can check that the reduction in total
freight cost has in fact been 20 cents per ton times the 10 tons diverted to the new
route.
Repeating the Process. The rest of the solution proceeds by mere repetition of the
process already followed for the first improvement in the program. We build up a
new cost table, Table G, by first copying from Table A the rates for the routes used
in Table F (these rates are shown in boldface type in Table G), then calculating the
35
10
10
5
35
25
25
MATHEMATICAL PROGRAMMING INFORMATION FOR DECISION MAKING
97
row and column values, and then filling in the other squares (lightface type). We next
compare Table G with Table A square by square and find that the square with the
largest difference in favor of G is D I (1.05 against 1.00). We therefore put fx in
D I of Table H, remove the "isolated" squares with stars, and then follow around a
circuit with +x and — x as indicated. The square with the smallest number with a
—x beside it is A I, with a value of 5, and we therefore add or subtract 5 as indicated
by +x or — x to produce Table J.
TABLE
G.
COSTS
FOR ROUTES USED
{Dollars per ton)
IN TABLE F
Factory
I
II
III
Ro
w value
Warehouse A
B
C
D
Column value
1.05
1.35
1.15
1.05
1.05
.90
1.20
1.00
.90
.90
1.10
1.40
1.20
1.10
1.10
.00
.30
.10
.00
TABLE
H
CHANGES TO BE MADE
(Tons)
IN TABLE F
Factory
I
//
III
Total
Warehouse A
B
C
D
5x
+x
5
30+x
30x
10*
5+x
25x
35
10
35
25
Total
60
40
105
TABLE J.
SECOND
REVISED PROGRAM
(Tons)
OF SHIPMENTS
Factory
/
II
III
Total
Warehouse A
B
C
D
Total
5
7
35
25
60
10
10
20
40
35
10
35
25
105
TABLE
K.
COSTS FOR ROUTES USED IN TABLE J
{Dollars per ton)
Factory
I
//
III
R<
ow value
Warehouse A
B
C
D
Column value
1.00
1.30
1.10
1.00
1.00
.90
1.20
1.00
.90
.90
1.10
1.40
1.20
1.10
1.10
.00
.30
.10
.00
From Table J we make up a new cost table, Table K. Comparing Table K with
Table A, we find that every lightface figure in Table K is smaller than the corre
sponding figure in Table A. There is no further improvement that can be made; in
fact, any change made in the program of Table J would result in an increase in the
cost of freight. Had there been squares where the lightface figure in Table K was
just equal to the rate in Table A, this would have indicated a route which could be
used without either raising or lowering the total cost of freight.
98 PLANNING AND CONTROLLING PRODUCTION LEVELS
Why the Procedure Works. To see why this method works, consider Table B. Now
suppose that we ship x tons from Factory III to Warehouse B. Every ton that we ship
will cost $1.40, the rate between these two points. But for every ton which B gets
from III, one less ton from II will be needed, thereby saving $1.40 of freight. Factory
III, on the other hand, cannot now supply both C and D as before, whereas Factory
II now has an excess. The simplest solution is to have III ship less to C, thus saving
$1.20 per ton, while II makes up the deficit at a freight cost of $1.00 per ton. The
net effect is a saving of 20 cents per ton, even though the shipments from III to B
cost just as much as the previous shipments from II to B.
This saving of 20 cents per ton is exactly the difference between the $1.60 in Square
B III of Table D and the $1.40 in the same square of Table A. This is true in gen
eral; the lightface figures in a "cost table" show the net savings on other routes which
can be made by readjusting the program if direct shipments are made along the
route in question. In other words, the lightface figures show the cost of "not using" a
route; the cost of using the route is, of course, simply the freight rate as shown in
Table A.
The best possible program has not been reached until there is no unused route for
which the cost of "using" is less than the cost of "not using." To be sure, at any stage
in the process of arriving at a best program there may be more than one route for
which the cost of not using is higher than the cost of using. We have given the rule of
making the change by introducing the route for which the difference between the two
costs is greatest. This rule is not necessary, but it is commonly believed that use of
this rule will usually reduce the number of steps required to arrive at the best possible
program.
Any program is a best possible program if there is no unused route for which the
cost of using is less than the cost of not using. This is a rather important fact, since it
means that a solution can be checked by simply building up the corresponding cost
table. There is no need to check over the work which produced the solution. Further
more, if there is an error in the solution, it is a waste of time to go back to find it;
everything will come out all right if you simply go on making successive changes until
the best possible program emerges. This is an additional reason why the transportation
problem procedure is really suited for hand computation while the general procedure
is not; there is a reasonably simple check on the accuracy of the final solution ob
tained by the general procedure, but correction of any errors that may be present is
far more difficult.
The map also shows why we arrived at the value 10 for the x in Table E. If we
make direci shipments from III to B, we must reduce shipments from II to B and
from III to C. We cannot reduce either of these below zero. The route from II to B
carries the smaller traffic, 10 tons, and therefore 10 tons is the largest amount we can
ship from III to B. Table E has — x beside each route that will be reduced as a result
of the change, and a +x beside each route that will be increased. The routes which
are starred in Table E are the routes which are not in the "circuit" IIIBIIC— III.
In some cases adjustments could be made which would give a greater saving per
ton or make possible diversion of more tons than will result from the use of the
rules given above. It is perfectly permissible to make more general changes in the
program at any stage provided that they are made in accordance with the rule given
previously for starting the program. On the other hand, such general adjustments are
never necessary, since it is absolutely certain that the stepatatime method described
above will ultimately lead to the best possible program.
Coping with Degeneracy. The procedure just described serves to solve any "trans
portation" problem of any size except when degeneracy appears in a route table at
some stage in the solution of the problem.
A route table is degenerate if it can be divided into two or more parts each of
which contains a group of factories whose combined capacity exactly satisfies the
combined requirements of the warehouses assigned to them. Table L gives an ex
MATHEMATICAL PROGRAMMING INFORMATION FOR DECISION MAKING 99
ample of such a situation which might have arisen in solving the example we have
just worked out. Warehouses A and D exactly use up the capacity of Factory II, while
Warehouses B and C exactly use up the capacity of Factories I and III. Under such
circumstances the procedure breaks down because it is impossible to build up the
cost table corresponding to a degenerate route table; that is, in this instance, the cost
table corresponding to Table L.
TABLE L. PROGRAM OF SHIPMENTS WHICH MIGHT HAVE
OCCURRED BEFORE REACHING SOLUTION
Factory I II III Tota i
Warehouse A 35 35
B 5 5 10
C 35 35
D 25 25
Total 5 60 40 105
The following simple device will take care of this difficulty: If the number of
plants is smaller than the number of warehouses, divide one unit of shipment by
twice the number of plants. (If shipments are to be measured to the tenth of a ton,
for example, we divide Mo ton, not 1 ton, by twice the number of plants.) Take any
convenient number which is smaller than this quotient and add it to the capacity of
each of the plants; add the same total amount to the capacity of any one warehouse.
If the number of warehouses is less than the number of plants, then reverse the rule.
In either case, solve the problem as if the additional quantities were real parts of
the requirements and capacities; then when the problem has been solved, round all
numbers containing fractions to the nearest unit of shipment. (A route carrying less
than onehalf unit is rounded to zero.) The solution thus obtained is not approximate;
it is exact.
When to Use
In its original application, as illustrated in the example worked through above, the
transportation problem consists of assigning a set of sources to a set of destinations in
such a way that the total cost of transportation from sources to destinations will be a
minimum. The capacity of each individual source and the requirements of each in
dividual destination are fixed in advance, and the total capacity equals the total re
quirements. A unit of requirements at any destination can be filled by the use of a
unit of capacity at any source, and only the cost of freight varies according to which
particular source is used.
This can easily be generalized as a problem of assigning a set of inputs of any
nature whatever to a set of outputs of any nature whatever in such a way that the
total cost of conversion is a minimum. The inputs might be the available supplies of
various raw materials, for example, rather than the capacities of various factories,
while the outputs might be the quantities produced of various products rather than
the quantities of a single product shipped to various warehouses.
There is no real change when the problem is one of maximizing profits rather than
minimizing costs. Instead of a "rate table" giving the cost of converting one unit of
any input into one unit of any output, we have a "margin table" giving the margin
which will be realized by such conversion, the margin being the revenue from selling
the unit of output less the variable costs of producing it. The program is developed in
exactly the same way as in the example worked through above, except that new
"routes" are introduced when the margin from not using the route is less than the
margin from using it, rather than when the cost of not using it is higher than the cost
of using it.
The formal characteristics which a problem must have if it is to be solved by the
transportation procedure are the following:
100 PLANNING AND CONTROLLING PRODUCTION LEVELS
( 1 ) One unit of any input can be used to produce one unit of any output.
(2) The cost or margin which will result from conversion of one unit of a
particular input into one unit of a particular output can be expressed by a single
figure regardless of the number of units converted.
(3) The quantity of each individual input and output is fixed in advance, and
the total of the inputs equals the total of the outputs.
If a problem cannot be put into the form specified by these three characteristics, it
cannot be solved by the transportation procedure. However, these are formal charac
teristics, and it is often possible to find devices or tricks which will put a problem
into this form even though at first glance it seems quite different. It is impossible to
give a complete list of such devices, but we shall describe here the more common
ones, which make it possible to solve by the transportation procedure all the prob
lems discussed on pages 5973.
Inputs and Outputs Not Fixed in Advance. In many problems all that we know in
advance is how much of a given input is available or how much of a given output
could be sold. We wish the program to determine how much of each it will be
profitable to use or make. This violates the third requirement stated above, but the
difficulty is easily overcome by the introduction of "dummy" inputs and outputs.
If, for example, total factory capacity exceeds total warehouse requirements, we
create a dummy warehouse and treat it exactly as if it were real. The cost or profit
which will result from supplying a unit to the dummy warehouse from any factory is
set down in the rate table as zero, and the requirements of the dummy warehouse are
set equal to the difference between total capacity and total real requirements. That part
of any factory's capacity which the final program assigns to the dummy warehouse
is capacity which is actually to be left idle.
If total potential output exceeds total available input, we create a dummy input
equal to the difference between the two. The cost or margin resulting from supplying
a unit of output from the dummy input is set at zero in the cost or margin table;
where the final program calls for producing all or a part of some output from the
dummy input, that amount of this potential output is not really to be produced at all.
In a case such as that described in Part II under the heading Where to Sell, it is
possible that potential inputs may be left unused at the same time that potential
outputs are left unfilled. This calls for the use of both a dummy input and a dummy
output. Since neither the total amount of real inputs which will be used nor the total
amount of real outputs which will be produced is known until the program has been
computed, the quantity of the dummy input must be set equal to or greater than the
total of the potential real outputs, and the amount of the dummy output must be set
equal to or greater than the total of the potential real inputs. With this proviso, the
quantities assigned to the dummies are arbitrary, except that the total of the real
plus dummy mputs must equal the total of the real plus dummy oz//puts. The final
program will show a certain amount of dummy output to be supplied from the
dummy input, but this figure has no real meaning whatever and should be disregarded.
Inputs and Outputs at Varying Prices. It may be that a factory can supply a certain
amount of product at one cost and an additional amount at a higher cost (for ex
ample, by the use of overtime), or that a certain amount of a material can be ob
tained at one price and additional quantities at higher prices. Similarly it may be
possible to sell a certain amount of product at one price and additional amounts only
at lower prices. All such cases are handled by treating the input at each cost as a
separate input, or the output at each price as a separate output. In this way we can
still produce a cost or margin table which shows a single unchanging perunit cost or
margin for converting any particular input into any particular output.
Note that this method will not work if the price at which the entire output is sold
depends on the quantity sold. As pointed out in Part II under the heading Price,
Volume, and Profit, this is not a problem of linear programming.
MATHEMATICAL PROGRAMMING INFORMATION FOR DECISION MAKING
101
Impossible Processes. The first formal requirement set forth above demands that
one unit of any output be producible from one unit of any input. In some cases
particular inputoutput combinations may be completely or practically impossible.
For example, freight service uniting a particular factory with a particular warehouse
may be so poor that management will in no case permit its use, or it may be simply
impossible to make a particular product from a particular material. This situation
causes no difficulty at all in the solution of the problem, since all we need to do is to
assign a fictitious, extremely high "cost" to the conversion of this input into this
output. In this way we can be sure that the unwanted process will not appear in the
final solution.
Artificial Units. In other problems, the amount of output which can be obtained
from a unit of input depends on the particular output and input in question. In prob
lems involving the selective use of raw materials, for example, the yield of any
material may depend on the product, and the amount of material required for a
particular product may depend on which material is used. Usually such problems
cannot be solved by the transportation procedure, but in some cases the data can be
reduced to such a form that they can.
This was true in the first rawmaterial problem discussed above. The trick here was
to express each output not in terms of the quantity of product but in terms of the
amount of Grade I material which would be required to produce it, and to express
the inputs of Grade II and Grade HI material in terms of the amount of Grade I
material which they could replace. This made it necessary, of course, to make
corresponding changes in the perunit purchase cost of Grades II and III material and
in all preunit processing costs. Table M shows the form to which Exhibit V had to
be reduced before computing the program of Exhibit VI.
TABLE M.
MARGINS,
SALES POTENTIALS, AND
AVAILABILITIES
Quantity
Product
A
B
C
D
available
(equivalent
Material
Margin
per
equivalent ton
tons)
I at $48/ton
$ 17
$ 32
$ 4
$ 17
100
I at $72/ton
( 7)"
8
(20)*
( 7)*
100
II at $24/ton
19
24
12
14
100
II at $36/ton
1
6
( 6)*
( 4)*
100
IIIat$18/ton
15
24
16 t
( 5)*
150
III at $24/ton
5
14
6
(15)*
250
Potential sales
(equivalent
tons)
240
150
240
90
* Minus quantity.
\ Derivation for Product C and Grade III material at normal price. As shown by the yield table
(Exhibit v), 2.5 tons of III replace 1.5 tons of I, so that 1 ton of III = .6 equivalent tons. As shown
by the same table, 1.5 tons of I are required to produce 1 ton of C, so that 1 ton of C = 1.5 equiv
alent tons.
Material available: 250 tons, or .6 X 250 = 150 equivalent tons.
Sales potential: 160 tons, or 1.5 X 160 = 240 equivalent tons.
Product price: $135 per ton, or $135/1.5 = $90 per equivalent ton.
Processing cost: $66 per ton of product, or $66/1.5 = $44 per equivalent ton.
Material cost: $18 per ton, or $18/.6 = $30 per equivalent ton.
Margin: $90 (selling price) — $44 (processing cost) — $30 (material cost) = $16 per equivalent
ton.
The reason why the subsequent cases discussed above could not be solved by the
transportation procedure should now be clear. If the rawmaterial problem were
changed so that the inferiority in yield of the lowergrade materials varied from
product to product, it would no longer be possible to express these inputs in such a
way that one unit of any input could produce one unit of any output. In the
machineshop problems, the amount of time on one machine which could be re
placed by one hour on another varied according to the product and the process being
102 PLANNING AND CONTROLLING PRODUCTION LEVELS
used. The Avgas problem is still more complex, since a single unit of any output is
blended from several inputs.
Such are the problems which call for the use of the general procedure.
The General Procedure
"Simplex method" is the technical name for the general procedure. Actually there
are two slightly different versions. The original version 10 will really work well only
for rather small problems because of the way in which rounding errors build up from
step to step. Machine computation of large problems is better carried out by the
modified method of Charnes and Lemke. 11
The general procedure can be worked by hand with the aid of a desk calculator
when the number of variables is small, as in the examples discussed in the main text.
However, it requires the use of automatic computers in most practical problems ow
ing not to the difficulty but to the sheer quantity of arithmetic involved. Even the
simplified Avgas problem discussed above required several days of hand computation
to solve by the general procedure, while the answer to a problem with twice as many
blending stocks and twice as many end products could be obtained in an hour or lessr
on a good electronic computer.
There are still certain limitations on the size of problem which can be handled
on existing computers with existing codes of instructions, and some problems which
can be solved may cost too much time or money to be worth solving. In many cases,
nevertheless, skilled mathematical analysis of a very large problem will show that it
can be simplified or broken into manageable parts.
Some problems will undoubtedly remain intractable, but until many more practical
applications have been made, it will not really be known whether this will prove to
be a frequent obstacle or a very rare one. It should be remembered that rapid progress
is being made both in mathematical research 12 and in the design of computers and
computing codes. If business finds that it is important to solve problems of linear
programming, it seems almost certain that means will be found of solving the great
majority of the problems that occur.
V.
Production Scheduling by the
Transportation Method of
Linear Programming*
Edward H. Bowman
With fluctuating sales, a manufacturer must have fluctuating production, or
fluctuating inventory, or both. Penalties are associated with either type of
10 See A. Charnes, W. W. Cooper, and A. Henderson, An Introduction to Linear Program
ming (New York, John Wiley & Sons, Inc., 1953).
11 See Proceedings of the Association for Computing Machinery (Pittsburgh, Richard
Rimbach Associates, 1952), pp. 9798.
12 An important recent advance is to be found in A. Charnes and C. E. Lemke, "Com
putational Theory of Linear Programming I: The 'Bounded Variables' Problem," O.N.R.,
Research Memorandum No. 10 (Pittsburgh, Graduate School of Industrial Administration,
Carnegie Institute of Technology, 1954).
* From Operations Research, Vol. 4, No. 1 (1956), 100103. Reprinted by permis
sion of the author and the Operations Research Society of America.
PRODUCTION SCHEDULING
103
fluctuation. Several papers place this problem into a conventional linear
programming framework. This paper suggests that the same problem may
be placed into a transportationmethod framework and, further, that many
transportation problems may be extended to include multiple time periods
where this is meaningful. A generalized scheduling problem is placed here
into the standard form of the transportation table.
Many manufacturing firms have fluctuating sales patterns, particularly on
a seasonal basis. Fluctuations in sales can be accommodated by fluctuations
in production or fluctuations in inventory, or by some combination of the
two. Penalties are associated with either type of fluctuation. The problem
may be conceived as one of balancing production overtimetype costs with
inventorystorage costs to yield a minimum total of these costs.
This problem has been placed by several papers 1,2 into a conventional
linearprogramming framework which may be solved by the Simplex
method. 3  4
It is the purpose of this paper to suggest that the same problem may be
placed into a transportationmethod framework, and further, that many
"transportation problems" may be extended to include multiple time periods
where this is meaningful.
TABLE I (
UNIT COSTS OF "SHIPMENT
Sales periods (destination)
Total
*i ca
(1)
(2)
(3) ..
(n)
§ pac
Inventory (n) ^ ities
^ Inventory (0)
£ Regular ( 1 )
o Overtime (1)
~ Regular (2)
"§ Overtime (2)
*£ Regular (3)
^ Overtime (3)
.9
o
3 ...
o Regular (n)
pu Overtime (n)
Cr
Co
Ci
Cr+Ci
Co+Ci
Cr
Co
2d ..
C R +2d . .
C +2Ci . .
Cr + Cj . .
Co+Ci . .
Cr ..
Co ..
. (nl)Ci
. C R +(n2)Ci
. C +(n2)Ci
. C B +(n2)Ci
. C +(/i2)C,
. C B +(n—3)Ci
. C +(n3)Ci
Cr
Co
nCi /o
CR+nCt #1
Co+nCi Oi
C R +(n\)Ci R 2
Co+(n\)d 02
C R \(n2)Cj Rb
C +(n2)Ci Os
Cr+Ci Rn
Co + d On
Total
requirements
Si
s 2
1J3 »Jn
J (6)
In
Table I shows how the productionscheduling problem may be thrown
into the standard form for the transportation method and hence may be
solved by this method. 5  6 The values within the table represent the unit
cost of "shipment" from the row source to the column destination. In
effect, each type of production, regular or overtime, in each time period
is considered a source of supply or input. Each period's sales requirement
104 PLANNING AND CONTROLLING PRODUCTION LEVELS
is considered a destination or output. It is possible to compute the cost
of each possible shipment, which is a combination of production and
storage costs.
Certain cellroutes, marked by dashes, are forbidden since it is, of
course, not possible to produce in one period and sell (and deliver) the
unit in a previous period. The conventional procedure can be used of
assigning a cost M to these cellroutes, representing a very high cost, so
that, in effect, economic considerations drive out of the solution a "ship
ment" that is impossible. Though the notation does not indicate it, the
production and storage costs for each assignment or cellroute may be
unique, i.e., production and storage costs need not be the same for dif
ferent time periods.
(0) Notation:
h = inventory at the end of the /th time period.
R i — maximum number of units which can be produced during /th time period
on regular time.
Oi = maximum number of units which can be produced during /th time period
on overtime.
Si = number of units of finished product to be sold (delivered) during /th time
period.
C R = cost of production per unit on regular time.
Co = cost of production per unit on overtime.
Ci = cost of storage per unit per time period.
<6) Slack total = / + 2i? + 2O25/„.
ADVANTAGES
The computational advantages of the transportation method of linear
programming are fairly well known. Practically, a further advantage is
that organizations with such a scheduling problem may have some facility
with the transportation method, and not with one such as the Simplex
method.
The method can be readily extended to several products. Each product
in each sales period becomes a separate destination (or column). Sources
would remain the same. Units of sales and production would, then, probably
be given in hours or other time units, and the appropriate cost of produc
tion and storage of an hour's output would be used in the computation.
Possibly the most important point to be made here is that many diverse
problems now taking the transportation framework may be extended to
include time periods. For instance, where a number of plants supply a
number of warehouses with given requirements, capacities, and costs, some
routings become advantageous (as revealed by the conventional transporta
tion method). However, the sales requirements of many firms fluctuate,
possibly to different extents in different areas and for different warehouses.
The question arises as to whether it is better to use some of the less costly
"routes" during the slack periods and thus incur the associated storage
PRODUCTION SCHEDULING 105
costs, or to use the more costly routes during the peak periods and so save
those storage costs. The same general methodology presented here can
be used in such a problem. Each plant in each time period becomes a
source, and each warehouse in each time period becomes a destination.
The appropriate storage or timeperiod costs are added to the conventional
manufacturing and shipping costs in the cost matrix. This problem can
also be further extended to more than one product.
A class at M.I.T.'s School of Industrial Management was given a
relatively simple scheduling problem to solve by the general Simplex
method, and then by the transportation method. The first solution took
two to three hours, the second fifteen to thirty minutes. Among other
reasons, this was due to the fact that in a problem of this type (with almost
half of the "routes" forbidden because of impossible time sequences) short
cuts and partial solutions by inspection become possible. 7 Though it is
felt that the method has merit, it is quite difficult to get meaningful costs
(linearity?), capacities (homogeneity?), and requirements (certainty?) Re
search involving the application of this model to several companies in the
Boston area is currently underway.
References
( 1 ) J. F. Magee, Studies in Operations Research 1: Application of Linear Pro
gramming to Production Scheduling, Arthur D. Little, Inc., Cambridge,
Massachusetts.
(2) Joseph O. Harrison, Jr., "Linear Programming and Operations Research,"
J. F. McCloskey and F. N. Trefethen (eds.) Operations Research for
Management, pp. 23133, The Johns Hopkins Press, Baltimore, 1954.
(3) G. B. Dantzig, "Maximization of a Linear Function of Variables Subject
to Linear Inequalities," T. C. Koopmans (ed.) Activity Analysis of Pro
duction and Allocation, Cowles Commission Monograph 13, Chap. XXI,
Wiley, New York, 1951.
(4) A. Charnes, W. W. Cooper, and A. Henderson, An Introduction to
Linear Programming, Wiley, New York, 1953.
(5) A. Charnes and W. W. Cooper, "The Stepping Stone Method of Ex
plaining Linear Programming Calculations in Transportation Problems,"
Management Science 1, 4969 (October, 1954).
(6) A. Henderson and R. Schlaifer, "Mathematical Programming, Better
Information for Better Decision Making," Harvard Business Review 32,
3 (MayJune, 1954).
(7) H. S. Houthakker, "On the Numerical Solution of the Transportation
Problem," /. Opns. Res. Soc. Am. 3, 210 (1955).
Chapter 5
PRODUCTION CONTROL WITH
LINEAR DECISION RULES
VI.
Mathematics for Production
Scheduling*
Melvin Anshen, Charles C Holt, Franco Modigliani,
John F. Muth, and Herbert A. Simon f
Fluctuations in customers' orders create difficult problems for managers
responsible for scheduling production and employment. Changes in ship
ments must be absorbed by some combination of the following actions :
Adjusting the amount of overtime work.
Adjusting the size of the work force.
Adjusting the finished goods inventory.
Adjusting the order backlog.
Since each of these courses of action has certain associated costs, one
of the prime responsibilities of production management is to make de
cisions that represent minimum cost choices. Difficult enough when fluctua
tions in orders can be predicted, the decisionmaking assignment is even
* From the Harvard Business Review, Vol. 36, No. 2 (1958), 5158. Reprinted by
permission of the Harvard Business Review.
t Authors' note: The research for this article was a group study carried out by mem
bers of the Office of Naval Research Project on Planning and Control of Industrial
Operations at the Carnegie Institute of Technology.
106
MATHEMATICS FOR PRODUCTION SCHEDULING 107
more complex in the common circumstance of unforeseen changes in de
mand. But the importance of the fundamental responsibility is clear. Better
decisions within a company contribute directly to its profits. Even more
broadly, better decisions in many companies can increase the efficiency
with which the nation uses its resources — a fact of growing importance as
our arms race with the Soviet Union intensifies.
This article reports some of the findings of a research team that has been
studying the application of mathematical techniques to the scheduling of
production and employment. As a result of this work, new methods have
been developed for improving the quality of scheduling decisions and for
helping managers to make substantially better decisions than they could
make by using prevailing ruleofthumb and judgment procedures. Once a
general rule has been developed, the computations required to establish a
monthly production schedule can be completed by a clerk in a few hours
or on a computer in a few minutes.
In a paint manufacturing plant the new methods were applied with
significant results :
A comparison of the actual performance of the factory under management's
scheduling decisions with the performance that would have been realized if
the new technique had been used indicated a cost advantage of at least 8.5%
for the mathematical decision rule, with further gains to be derived from im
proved sales forecasting. The plant was not a large one; there were only 100
employees. Yet the annual saving amounted to $51,000, reflecting reductions
in a number of cost items, including regular payroll, overtime, hiring, training,
layoff, and inventory.
The specific decision method described in this article is applicable to
other plants with similar production flows and cost structures. Moreover,
the general mathematical method can be adapted to production scheduling
in plants with different cost structures. Ultimately, the basic technique
should be applicable in areas other than production scheduling.
PRODUCTION AND MATHEMATICS
To use mathematics as a tool, one must understand it as a language.
Since it differs from the language of production, the essential first step
in applying it to a plant problem is to translate the description of produc
tion from its familiar vocabulary into the language of mathematics.
Such a transformation calls for generalizing, quantifying, and identifying
the goals and constraints (limitations or restrictions). Data drawn from
financial and cost accounting systems are useful for this purpose, but they
are not ordinarily sufficient. They need to be supplemented by quantitative
108 PLANNING AND CONTROLLING PRODUCTION LEVELS
approximations of production functions that are seldom described nu
merically. This may call for simplification and aggregation. Fortunately, as
the following comments demonstrate, the actual transformation is less
formidable than these words may suggest.
ELEMENTS OF DECISION
At one end of the production process, orders (on hand or anticipated)
generate production. At the other end of the process, shipments satisfy
orders. Within these limits the process accumulates costs.
Total costs for a given time period are influenced by managerial schedul
ing decisions. These decisions commonly are taken with reference to selected
goals. Certain costs, for example, are associated with the stability of the
production schedule over time:
If there is steady employment of a group of workers, costs are lower than
if the group fluctuates in size. Costs associated with hiring, training, layoff,
and overtime are minimized, as well as the less tangible costs related to under
time operations.
If incoming orders are not stable, however, a level rate of production can be
maintained only by accepting fluctuations in the order backlog or by making
shipments as required from a buffer stock of finished goods.
A decision to absorb fluctuations through finishedgoods inventory commits
the firm to direct investment costs and to the expenses associated with storage,
handling, spoilage, obsolescence, and adverse price changes. Similarly, a deci
sion to absorb order fluctuations through a buffer backlog also has recogniz
able costs associated with it, although these are not measured by standard
accounting techniques — the costs of customer dissatisfaction, loss of future busi
ness, and adverse price changes.
In most work settings, production decisions are further complicated by
the movement of several products through common facilities and work
groups. Another problem often encountered is the variable procurement
costs for materials and parts, which are related to purchase lot size and
stability of incoming deliveries.
Finally, decision strategy must consider the effect of errors in forecasting
future orders and of the accumulation of scheduling decisions over succes
sive time intervals. Both these considerations compel the adoption of a
dynamic strategy that combines a judgment as to the impact of the
ordersstockproductionshipment complex on the immediately upcoming
time period with a judgment designed to compensate for prior errors in
scheduling for preceding time periods.
MATHEMATICS FOR PRODUCTION SCHEDULING 109
GETTING OPTIMUM RESULTS
The best, or minimumcost, decision in this complicated setting with its
multitude of interrelated variables is far from obvious. There is no easy way
out : One management may pursue a shifting strategy outlined by ruleof
thumb procedures; another management may adopt a stable strategy de
signed to realize a single objective, such as level employment or prompt
delivery of customers' orders. However, little argument is needed to show
that such strategies cannot, except in extraordinary circumstances, produce
optimum results.
Since every fluctuation in incoming orders can be met only by a choice
among alternatives, each of which carries an inescapable set of associated
costs, the scheduler is confronted with a complex and dynamic situation
in which optimum performance requires absorption of fluctuations through
a carefully weighted allocation among all buffer elements. Part of the im
pact may be taken by inventory adjustments (in both order backlogs and
finished goods), part by overtime and undertime scheduling of workers,
and part by changes in the size of the work force.
The best mix of these elements clearly depends on the nature of the
production process (for example, the feasibility of smooth rather than
onestepatatime adjustments in the scale of operations, the relation of
setup costs to length of run, and so on), and the cost structure in an in
dividual plant. Even for a specific plant, the optimum allocation will
change with the frequency, amplitude, and predictability of fluctuations
in orders.
DESCRIBING THE PROBLEM
The mathematical approach to any decisionmaking problem requires
several distinct steps:
(1) Managers must agree on the objective of maximizing or minimizing a
specific criterion. For the firm as a whole this criterion would be profits. For the
productionscheduling manager who controls neither sales nor profits, the criti
cal criterion would be minimizing the costs of operations.
(2) All costs must be described quantitatively in comparable units, including
intangible costs and those not regularly identified by financial and costaccount
ing systems.
(3) A reporting and planning period must be selected for the accumulation
and analysis of information relevant to scheduling decisions. The selection of
the decision period is itself a problem. The significant factors include the size of
errors in forecasting incoming orders, the cost of making forecasts, the time
required to gather new information to improve earlier forecasts, the cost of
making and administering decisions, and the relative costs of making a large
number of small scheduling changes and a small number of large ones.
110 PLANNING AND CONTROLLING PRODUCTION LEVELS
The process of quantifying intangibles, such as the costs associated with
maintaining a buffer backlog of accepted but unproduced orders, is a
process of making numerically explicit certain values that are always present
in management thinking but in an illdefined and cloudy form. Actually,
precision in doing this is neither possible nor necessary. But it is essential
to assign numerical weights to all variables and to recognize that doing this
is no more than translating from a language that permits the implicit
to a language that compels the explicit.
Further, the general decision problem must be expressed in a mathe
matical form that is flexible enough to comprehend the full range of
production costs and simple enough to permit relatively easy solution. If
we consider the nature of the costs associated with production, as outlined
above, we will find that a Ushaped curve is a useful general expression.
For example:
High costs are incurred in holding both large inventories and inventories so
small that outofstock conditions are common, with consequent delays in ship
ments, short production runs to fill back orders, and customer dissatisfaction.
Similarly, frequent scheduling of both overtime and undertime (a less than
fully employed work force) is expensive. Such costs are often regarded as in
tangible and are not explicitly reported in accounting procedures; but they
must be explicitly quantified for mathematical treatment. Somewhere between
the extremes of overtime and undertime, labor costs are at a minimum.
These considerations indicate the feasibility of achieving a reasonable
and workable approximation of the complex of production costs by the
simplest mathematical expression that gives a Ushaped curve — a quadratic
function.
ASSUMPTIONS AND ADVANTAGES
It should be observed that the mathematical view of the problem does
not assume that the costs of hiring workers equal the costs of laying them
off, or that changes in costs in either direction are symmetrical. It does
not assume that the costs associated with adding to inventory holdings
equal the costs of depleting inventory, or that changes in either direction
are symmetrical.
One common misunderstanding about the language of mathematics is the
belief that precise numerical expression requires equal precision in report
ing "facts." Mathematics can be an effective decisionmaking tool even in
circumstances in which the values assigned to costs represent no more
than approximations.
In this sense, the mathematical approach is more precise and consistent,
and therefore more rational, than judgment based on experience and in
MATHEMATICS FOR PRODUCTION SCHEDULING 111
formed hunch. It compels the scheduler to consider all criteria previously
defined as essential, and it compels him to consider them consistently every
time a scheduling decision is made. In fact, after the decision rule, expressed
as a formula with explicit values for specified constant elements, has been
framed, it does the considering for the scheduler as a routine of the mathe
matical process.
It follows that the ultimate judgment of the efficacy of a mathematical
decisionmaking process in a production setting is not in terms of its ability
to schedule for minimum true costs. After all, the truth about all costs
probably can never be determined. But it can be demonstrated mathe
matically that the decisions arrived at by means of the rule are the optimum
decisions for the assigned cost values.
The important test thereafter is showing that the decisions arrived at by
means of the mathematical tool are better decisions, and that operations
are scheduled at lower costs, than decisions arrived at by alternative
methods. This can be demonstrated by matching the actual record under
established scheduling procedures with the record that would have been
made under the mathematical decision rule.
NEW METHODS APPLIED
To test the application of the general mathematical techniques described
above, the research team studied the scheduling problem in a paint factory.
To simplify the analysis, without changing its fundamental concept, schedul
ing decisions were assumed to be made monthly and costs were accumulated
over the same period.
COST COMPONENTS
First the following kinds of cost components were identified:
Regular payroll, hiring, and layoff.
Overtime and undertime.
Inventory, back order, and machine setup.
These costs were developed as discrete components and then were com
bined in an expression of the complete cost function for the factory as a
whole.
Payroll. With monthly adjustments in the size of the work force, regular pay
roll costs per month were a linear function of the size of the work force; that is,
if they had been correlated on a graph, with payroll costs on the vertical axis
and workforce size on the horizontal axis, the resultant diagonal line would
have been fairly straight. Payroll dollars for regular work time also varied
directly with the size of the work force measured in manmonths.
112 PLANNING AND CONTROLLING PRODUCTION LEVELS
In contrast, hiring and layoff costs were associated with the magnitude of
change in the size of the force. Costs of hiring and training rise with the number
of workers hired and trained; layoff costs are associated with the number of
workers discharged. There is no necessary symmetry between hiring and layoff
costs in their relation to the number of workers processed; and random factors,
reflecting the tightness of the local labor market or reorganization of the work
structure at certain levels of employment, may also be present sporadically.
The representation of these costs by a Ushaped curve, therefore, was only an
approximation of the average costs of changes of various magnitudes in the
work force.
Overtime. Overtime operations in the factory involved wage payments at an
hourly rate 50% higher than the regular time rate. Undertime costs, reported
only indirectly through the accounting system, reflected waste of labor time
measured by the difference between the actual monthly wage bill and the wage
bill for the smaller work force that would have sufficed to accomplish the actual
production. Actual overtime during any month is determined, of course, not
only by a work load in excess of that which can be produced by the regular
force in regular hours, but also by such random disturbances as emergency
orders, machine breakdown, quality control problems, fluctuations in produc
tivity, and so on.
In setting the production rate and the work force for a month, the scheduler
must balance the risk of maintaining too large a work force against the risk of
holding a smaller work force but being required to pay overtime compensation.
As in the case of hiring and layoff costs, these considerations suggested a
Ushaped, possibly unsymmetrical, cost curve.
Inventory. Absorbing order fluctuations through inventory and backorder
buffers gives rise to new costs. Holding a goodsize inventory incurs costs such
as interest, obsolescence, handling, storage, and adverse price movements. On
the other hand, a decision to reduce these costs by operating with a smaller
inventory invites outofstock conditions with the associated costs of delayed
shipments, lost sales, and added machine setups for special production runs to
balance out stocks and to service mandatory shipments. The analysis pointed to
the need for an optimum inventory level at which combined costs were at a
minimum.
COST FUNCTION DEVELOPED
The complete cost function for production and employment scheduling
was developed by adding the components reviewed above. (For its mathe
matical form see Appendix, Reference 1.) The mathematical generaliza
tion was then applied to the specific situation in the paint factory by insert
ing numerical values representing estimates of the various costs involved.
Some of the estimates were drawn directly from accounting data or ob
tained through statistical treatment of accounting data. Other estimates,
such as those for the intangible costs of delayed shipments, were subjective.
MATHEMATICS FOR PRODUCTION SCHEDULING
113
Here it is important to note that the accuracy of the estimates was not a
critical consideration. An analysis of the effect of errors as large as a factor
of two — that is, overestimating specific cost elements by 100% or under
estimating them by 50% — indicated that use of the resultant decision rules
would incur costs only 11% higher than with correct estimates of costs.
DECISION RULES
At this point the mathematical process led to the development of two
monthly decision rules, one to set the aggregate rate of production and
the other to establish the size of the work force. (For the mathematical
derivation of these rules see Appendix, Reference 2.) The two rules are
set forth in Exhibit I.
EXHIBIT I.
Pt
+ .993W t i+ 153. .4641,
Wt = .743 Wti + 2.09  .010 I t i +
PRODUCTION AND EMPLOYMENT DECISION RULES
FOR PAINT FACTORY
+.463 O t
+.234 O t+1
+ .111 Ot + 2
+.046 O t+3
+.013 O t+ 4
—.002 Ot +5
—.008 Ot + e
.010 O t+ 7
.009 O t+8
.008 Ot +e
.007 O t+ io
.005 Ot + n
+ .0101 O t
+.0088 O t+ i
+.0071 O t+2
+.0054 O t+3
+.0042 O t+i
+.0031 O t+5
+.0023 Ot +6
+.0016 Ot +7
+.0012 Ot +8
+.0009 O t+e
+.0006 O t+ io
+.0005 Ot + n
Where:
Pt is the number of units of product that should be produced during the forthcoming
month, t.
Wti is the number of employees in the work force at the beginning of the month (end
of the previous month).
Iti is the number of units of inventory minus the number of units on back order at
the beginning of the month.
W t is the number of employees that will be required for the current month, t. The
number of employees that should be hired is therefore Wt — Wti,
O t is the forecast of number of units of product that will be ordered for shipment dur
ing the current month, t.
Ot+i is the same for the next month, t+ 1 ; and so forth.
The production rule incorporates a weighted average of the forecasts
of twelve months' future orders, which contributes to smoothing production.
The weights assigned to future orders decline rapidly because it is not
114 PLANNING AND CONTROLLING PRODUCTION LEVELS
economical to produce for distant shipment in view of the cumulative cost
of holding inventory. (This accounts for the negative numbers for the last
seven months in the production rule in Exhibit I.) The employment rule
also incorporates a weighted average of forecasts of future orders, with
the weights projected further into the future before becoming negligible.
The second term of the production equation (.993 W*_i) reflects the
influence of the number of workers employed at the end of the preceding
month. Because both large decreases in the payroll and large amounts of
unused labor are costly, the level of scheduled production responds to the
size of the work force at the start of the month.
The next two terms in the production decision rule (513. — .464 I*_i)
relate the inventory to production. If net inventory at the end of the pre
ceding month is large, the negative term will exceed the positive term, with
a resultant downward influence on scheduled production. A reverse rela
tionship would contribute to establishing a higher level of production. This
term also functions to take account of past forecast errors, since their
effect is to raise the net inventory above, or push it below, the desired level.
The first term of the employment rule (.743 W f _i) provides for a
direct influence between the work force on hand at the beginning of a
month and the scheduled employment during the month, reflecting the costs
associated with changing the size of the work force. The next two terms
^^f2.09 — .010 l t i) make provision for the effect of the net inventory
position on the employment decision. A large net inventory will lead to a
decrease in the scheduled work force, and a small net inventory will have
the opposite effect.
The terms of the two rules make explicit the dynamic interaction of pro
duction and employment. For example, production during a month af
fects the inventory position at the end of the month. This affects the
employment decision in the next month, which then influences the produc
tion decision in the third month. Again, the influence of net inventory on
both production and employment decisions provides a selfcorrecting
force which operates to return inventory to its optimum position regardless
of the accuracy of sales forecasts.
It is most difficult, if not impossible, to account for this interaction with
out a mathematical decision rule. The manager who makes these decisions
on the basis of intution and experience may hit the right answer some of the
time, but he will not do so consistently.
The weighting of the sales forecasts and the feedback factors determines
the magnitude of production and employment responses to fluctuations in
orders, thereby allocating the fluctuations among work force, overtime, in
ventory, and backlog in the interest of minimizing total costs. While the
work force responds to rather longrun fluctuations in orders, the principal
MATHEMATICS FOR PRODUCTION SCHEDULING 115
response of production is to nearterm orders and to the inventory position.
Thus, the rule provides for the absorption of shortrun fluctuations in orders
and errors of forecasting by scheduling overtime and undertime operations.
SUPERIORITY OF DECISION RULES
How much are decision rules of the kind described an improvement over
the usual methods of scheduling production?
This question was answered for the paint factory by making a hypo
thetical application to scheduling in the plant and comparing the results
with actual performance under established procedures. Production and
employment decisions in the paint factory were analyzed for a sixyear
period. The production and employment decision rules were then applied
to simulate the decisions that would have been made if they had been
in use during the same sixyear period.
Because the same data were used by the research team as by manage
ment, hindsight could be of no advantage except in one situation, and here
measures were taken to counteract it. A necessary ingredient for the com
parison was a monthly series of forecasts of future orders throughout the
period under analysis. Because no such forecasts had actually been recorded,
the comparison could not be made on the basis of forecasts identical to
those implicitly in the minds of management when it made its scheduling
decisions. As a substitute, two sets of forecasts were devised which
bracketed the forecasts actually used by management:
The first set of forecasts consisted of actual orders received. This was, in other
words, a "perfect" forecast, assuming the future to be known in the present; use
of it established an upper limit for performance.
The second set of forecasts was derived by assuming that future orders would
be predicted by a moving average of past orders. Specifically, orders for a year
ahead were forecast as equal to those actually received in the preceding year.
This annual forecast was then converted to a monthly forecast by applying a
seasonal adjustment based on actual past performance.
A comparison of actual costs under management scheduling with hypo
thetical costs under the decision rules did not tell the whole story. The
figures were not solid; problems of allocating costs between paint and the
other products processed in the plant, as well as the absence of a firm
accounting underpinning for certain intangible costs, gave a tentative quality
to the data. The research team judged, however, that the comparison was
a valid one for all practical purposes and that the cost differences shown
in Exhibit II were highly significant. The figures cover two periods:
1. The longest period for which cost figures were available for a threeway
comparison between actual performance and expected performance under the
116 PLANNING AND CONTROLLING PRODUCTION LEVELS
new rules using both a perfect forecast and a movingaverage forecast, 1949
1953.
2. The period in which company performance was matched against the de
cision rule using a movingaverage forecast, 19521954.
Exhibit II shows that the general effect of the decision rules, with either
movingaverage or perfect forecasts, was to smooth the very sharp month
tomonth fluctuations in both production and size of work force in actual
factory performance. Overtime and inventoryholding costs were somewhat
higher under the rules with the movingaverage forcast (a "backward
looking" forecast) than the actual costs were, but this excess was more
EXHIBIT II. ACTUAL PERFORMANCE VS. EXPECTED PERFORMANCE
UNDER DECISION RULES
(In thousands of dollars)
Decision rule
Company
Movingaverage
Perfect
Costs
performance
forecast
forecast
A . Cost Comparisons
for 1949
1953
Regular payroll
$1,940
$1,834
$1,888
Overtime
196
296
167
Inventory
361
451
454
Back orders
1,566
616
400
Hiring and layoffs
22
25
20
Total cost
$4,085
$3,222
$2,929
139%
110%
100%
B. Cost Comparisons
for 1952
1954
Regular payroll
$1,256
$1,149
Overtime
82
95
Inventory
273
298
Back orders
326
246
Hiring and layoffs
16
12
Total cost
$1,953
$1,800
108.5%
100%
than offset by the fact that back orders were consistently held at lower
levels. It is worth observing that the costs associated with back orders are
particularly difficult to include as significant factors in ruleofthumb and
judgment decisions.
The decision rule with the movingaverage forecasts saved $173,000
annually against factory performance. For this stage in the history of this
plant, greater savings could have been secured by making optimum use
of crude forecasts than by improving forecasts. Note that the decision rule
with perfect forecasts had lower costs than the same rule with the moving
average forecasts in the 19491953 period — by 10%, or an average of
$59,000 annually. This difference, which is entirely attributable to better
MATHEMATICS FOR PRODUCTION SCHEDULING 117
forecasting, is a sizable one but only about a third as large as the other
saving.
In the 19521954 period actual factory costs exceeded costs under
the decision rule by 8.5%, or $51,000 per year on the average. The econ
omies of the decision rule were achieved by (a) reducing payroll costs
more than overtime costs increased, (b) reducing backorder penalty costs
more than inventoryholding costs increased, and (c) reducing hiring
and layoff costs.
CONCLUSIONS
While further exploration of the problems involved in applying mathe
matical decision rules to production and scheduling decisions seems clearly
desirable as a basis for definitive conclusions, the study reported in this
article provides firm support for several preliminary judgments. Empirical
experience with the rules in the paint factory corroborates the findings of
the research team. The methods have been in actual and satisfactory
operation in the factory for several years now, and their use is currently
being extended to other factories operated by the same company. The same
methods have also been adapted to several other productionscheduling
situations in other companies and have satisfactorily passed "dry run"
tests preliminary to actual installations in these situations. This report
of findings is confined to the paint factory study because this is the only
one for which the data are publicly available at the present time.
USE OF RULES
Decision rules supplement, rather than displace, management judgment
in scheduling production and employment. As such they are of great value
in helping management to:
1 . Quantify and use the intangibles which are always present in the background
of its thinking but which are incorporated only vaguely and sporadically in
scheduling decisions.
2. Make routine the comprehensive consideration of all factors relevant to
scheduling decisions, thereby inhibiting judgments based on incomplete,
obvious, or easily handled criteria.
3. Fit each scheduling decision into its appropriate place in the historical series
of decisions and, through the feedback mechanism incorporated in the de
cision rules, automatically correct for prior forecasting errors.
4. Free executives from routine decisionmaking activities, thereby giving them
greater freedom and opportunity for dealing with extraordinary situations.
In the case of the paint factory, for example, use of the decision rules
permits regular monthly scheduling of production and employment to be
118 PLANNING AND CONTROLLING PRODUCTION LEVELS
come a clerical function. Management attention can now be directed to
refining cost estimates and periodically adjusting estimates to reflect changes
in costs resulting from modifications of work flow and production process.
Beyond this, management has more time to consider nonroutine factors
and special situations that might provide reasons for modifying scheduling
decisions computed from the mathematical rules. Anticipated changes in
raw material availability, in the supply of workers with necessary skills, in
customers' procurement requirements, or in the character of competitors'
service offerings can get the attention they deserve from executives relieved
of the burden of repetitive, complex scheduling decisions.
Management time is also free to develop ways and means of improving
sales forecasting, with the knowledge that such gains can be fed directly
into the decision rules and thus improve their efficiency.
But it would be shortsighted to think of the decision rules only in terms
of the production setting of the paint factory. They can be modified to
apply to other types of scheduling problems. The required changes are in
the specific cost terms, not in the general structure of the rules. To be sure,
the development of the rules in a different kind of plant requires careful
study of the costs that are relevant to scheduling decisions, supported by
explicit quantification of all cost elements. Subject to this limitation, how
ever, the general technique is applicable to scheduling in any plant in
which the relevant costs may be approximated by Ushaped curves.
Decision problems in areas outside production would also appear to be
candidates for the application of mathematical decision rules of the type
described. The scheduling of warehouse operations, of employment in re
tail stores, of certain classes of retail merchandise stocks, of working
capital, and of some types of transportation operations — all appear to be
fruitful areas for research. And with ingenuity management will undoubt
edly discover still other applications in the future.
Appendix
The broad implications of this study should be of interest not only to
those persons directly concerned with production management but also to
a wide managerial group. A more detailed, technical presentation of this
research can be found in the following references:
(1) C. C. Holt, F. Modigliani, and H. A. Simon, "A Linear Decision Rule
for Production and Employment Scheduling," Management Science,
October 1955, p. 1.
(2) C. C. Holt, F. Modigliani, and J. F. Muth, "Derivation of a Linear De
cision Rule for Production and Employment," Management Science,
January 1956, p. 159.
LINEAR DECISION RULES AND FREIGHT YARD OPERATIONS 119
(3) H. A. Simon, C. C. Holt, and F. Modigliani, "Controlling Inventory and
Production in the Face of Uncertain Sales," National Convention Trans
actions, American Society for Quality Control, 1956, p. 371.
(4) H. A. Simon, "Dynamic Programming Under Uncertainty with a Quadratic
Criterion Function," Econometrica, Volume 24, p. 74.
VII.
Linear Decision Rules and
Freight Yard Operations*
Edwin Mansfield and Harold H. Wein]
The scheduling of output and employment is an important topic in In
dustrial Engineering and management science. Linear decision rules have
recently been devised to deal with this problem, and in practice they have
resulted in substantial savings. 1 In this paper we explore the usefulness of
such rules in scheduling output and employment in freight yards. The
results are tentative and subject to obvious limitations, but they seem to
indicate that these rules could be useful there. If so, linear decision rules
may provide at least a partial solution to an important railroad problem.
At the outset, we should note that the model used here is a first approxi
mation. Many factors that may be important are given only brief attention
and some aspects of our formulation of the problem are tentative. We men
tion these difficulties below, although we do not always take them up in
great detail.
* From The Journal of Industrial Engineering, Vol. 9, No. 2 (1958), 9398. Re
printed by permission of The Journal of Industrial Engineering.
t This report is based on research supported by a grant from the Westinghouse Air
Brake Corporation to Carnegie Institute of Technology. It is part of a larger project
concerning the railroad industry. We should like to acknowledge the valuable assist
ance of C. Link, R. Nadel, and E. Saunders of the cooperating railroad, and the com
ments of our colleague, J. Dreze on an earlier draft.
1 These rules were proposed by C. Holt, F. Modigliani, and H. Simon (3) and
C. Holt, F. Modigliani and J. Muth (4). For an earlier work, see C. Holt and H.
Simon (5); and for the proof of a basic proposition involved, see H. A. Simon (12).
120 PLANNING AND CONTROLLING PRODUCTION LEVELS
SCHEDULING OUTPUT AND EMPLOYMENT
IN FREIGHT YARDS
We begin this section by briefly describing 1. a freight yard, and 2. the
switching function. Then the scheduling problem is considered. Note at the
outset the importance of freight yards to the firm as a whole. The operation
of these yards may result in about onethird of a firm's total operating costs.
1. Freight yards differ with respect to size and layout, but they all contain
sets of tracks. In large yards, the track layout usually consists of a receiving
area where incoming cars are stored, a classification area where they are
switched, and an outbound area where they are stored before being hauled away
as a train.
2. Although it is not the only function of a yard, switching is certainly one
of the most important functions. Cars arrive on incoming trains and they must
be sorted out to form outgoing trains. This sorting operation is called switching.
It is performed by a switchengine that pulls a group of cars from a receiving
track and shoves the cars in the group onto the appropriate classification tracks. 2
Freight yards differ greatly with regard to the number of cars switched per day.
Some small yards switch fewer than 50 cars whereas a few very large yards may
switch as many as 4000 cars.
In this paper, we are concerned chiefly with large yards, i.e., those that
switch 1500 or more cars per day. At such yards, most engine crews are
specialized; certain crews do practically nothing but switching and no other
crews can engage in such work. We are interested in the scheduling of the
switching output and the switchengine crews. To fully understand this
problem, it is necessary to consider the conditions under which the yard
management makes decisions concerning output and employment and the
costs that must be considered when a decision is made.
The general yardmaster in a freight yard usually makes these decisions.
In doing so, he operates within a somewhat different framework than that
typically found in manufacturing. 1. There is no possibility of producing
for inventory. The railroads are a service industry and the yard can perform
services only on demand. 2. He cannot reject "orders." Under normal
circumstances, he must switch all cars that arrive. 3. The planning period is
generally quite short. Each day, the yardmaster decides how many switch
engine crews will be used during the next twentyfour hours and how they
will be allocated during the period. 3
2 For a more complete description of freight yard operations, see M. Beckmann,
C. McGuire, and C. Winston ( 1 ) .
3 It is commonly possible for extra crews to be hired or laid off up to two hours
before the beginning of a shift. Hence, the yardmaster can often change the number
of crews during one shift for the next. In practice, however, it seems that he usually
plans for the entire next day and that his plans are seldom altered appreciably. A
brief discussion of the problem concerning the proper planning period is contained
in a following section.
LINEAR DECISION RULES AND FREIGHT YARD OPERATIONS 121
When he plans the switching output and the number of switchengine
crews for the next day, the yardmaster is uncertain about the number of
cars that will arrive to be switched and the average productivity of the
crews. He must forecast these variables as best he can. Crew productivity
is generally treated as a constant unless climatic conditions are abnormal
or congestion exists in the yard. 4 In forecasting the number of car arrivals,
he often uses advance information concerning some types of traffic. 5 This
information is helpful, but it by no means dispels the problem of forecast
ing.
Having made these forecasts, the yardmaster is ready to decide on the
number of cars to be switched on the next day and the number of switch
engine crews to be used then. 6 The various strategies that are open to him,
and the costs that must be considered are discussed in the following section.
ALTERNATIVE STRATEGIES AND
RELEVANT COSTS
The yardmaster's decision on a given day is but one in a long sequence
of such decisions. A useful way of viewing his problem is to consider how
he should react to changes in the number of cars that arrive to be switched.
There are, in fact, large daytoday fluctutations in the number of car
arrivals, and he may adopt many strategies to meet them. 1. He may try
to vary the number of crews and the number of cars switched in accordance
with variations in the forecasted number of car arrivals. 2. He may try to
vary the number of cars switched in accordance with these variations; but
4 The productivity of the crews is relatively low when it is very cold, icy, etc. When
many unswitched cars accumulate in the yard, the latter becomes congested. Serious
congestion also reduces productivity because a larger part of a crew's time must be
devoted to "preparatory moves."
5 He derives this information mainly from teletype "consists" of symbol trains and
from conversations with the division chief dispatcher. The former provide him with
information concerning the number of cars on some important trains that will arrive
up to about 7 hours hence. The latter provide him with information concerning the
time at which trains will arrive.
6 A freight yard is a service installation, and the number of cars that the yardmaster
switches during the next day is not wholly under his control. In particular, he cannot
switch more cars than are available to be switched. In formulating the linear decision
rules, we neglect this constraint on the switching output. This seems to be legitimate
so long as the rules very seldom call for the constraint to be violated. In the freight
yard studied, the output prescribed by the rules violated this constraint on the follow
ing percentage of days: perfect forecast — 0%, naive forecast — 6%, yardmaster's fore
cast — 0%. On the basis of this evidence it appears that the prescribed output does
not violate the constraint so long as the forecasts are at all adequate. Of course, this
evidence pertains only to 61 days. Another problem here is the distribution of car ar
rivals over time. One may take this roughly into account by linking it up with crew
productivity. Still another problem revolves about the output measure used here. The
number of "cuts" switched might be used instead.
122 PLANNING AND CONTROLLING PRODUCTION LEVELS
he may maintain a constant number of crews, and use overtime to do the
necessary switching. 3. He may switch a constant number of cars, use a
constant number of crews, and allow the backlog of unswitched cars to
fluctuate freely. 4. He may use some mixture of the foregoing. 7
His choice among these strategies will depend on the costs associated
with them. Clearly, the costs that are relevant are straighttime crew costs,
overtime costs for crews, backlog costs, and costs associated with changes
in the number of crews. We shall proceed to describe these costs and to
represent them as functions of the decision variables. 8
The straighttime crew cost seems to be selfexplanatory. On the /th
day, it is proportional to the number of crews used then (Ki).
The overtime cost for crews needs no explanation. Given the number of
crews (Ki) and their average productivity (P), this cost is probably close
to proportional to Si — PKi where Si is the number of cars switched on the
/th day. For S t < PK it it is probably close to zero.
The backlog in a freight yard is the number of unswitched cars that are
present in the yard. 9 Backlog costs are the costs associated with various
backlogs. Two important costs that are included are the cost of car delay
and the cost of productivity decreases due to congestion. Congestion occurs
when many unswitched cars accumulate in the yard. As the congestion
becomes more severe, the productivity of the crews is reduced and the
costs of unsatisfactory service increase because more cars are delayed.
Three simplifications are made in treating backlog costs. 1. We make
them depend only on the size of the backlog. No account is taken of the
types of cars in the backlog. 2. We make them depend only on the backlog
at the end of the day. Although the backlog varies during each day, it does
not seen to depart very greatly from that at the end of the day. 3. We
make all costs accrue to the day when the backlog was formed. The costs
7 For a more detailed account of the ways in which a plant may adapt to fluctua
tions in demand, see (3, p. 35).
8 The decision variables are, of course, the number of switchengine crews and the
number of cars switched. The costs that are discussed do not include all yard costs.
Most of the excluded costs are essentially fixed or independent of the decision vari
ables. Inspection, oiling and related costs can be included in the analysis but, for
simplicity, they were omitted. The costs included here differ substantially from those
included in our other papers.
9 It is possible for a car to be switched more than once in a yard. In this case,
there may be some ambiguity in our definition of backlog. For example, a car has
been switched once but it must be switched again. Is it included in the backlog? We
did not include it. But if it seems important to do so, the analysis may very easily be
adapted accordingly.
LINEAR DECISION RULES AND FREIGHT YARD OPERATIONS 123
may actually be spread over subsequent days, but it is more convenient to
lump them together and charge them to the initial day. 10
The cost of various backlogs is quite small for backlogs of intermediate
size. 11 But outside this intermediate range, the costs rise steeply. The in
crease on the right is due to congestion. The increase on the left is due to
the fact that a portion of the backlog arrived just before the end of the
day. It would be very costly and sometimes impossible to switch these
cars before the end of the day. 12
Costs are also associated with changes in the number of crews. A sub
stantial increase in the number of crews can often be effected only by the
addition of less productive crews. A substantial decrease in the number of
crews is often costly because union agreements prescribe penalities for
laying off regular crews. The magnitude of these costs depends, of course,
on the caliber of the extra crews and the nature of local labor agreements.
At the freight yard studied below, these costs seem to be very small. But
at other yards, they are likely to be much more important. 13
LINEAR DECISION RULES
In planning the number of crews and the volume of switching for the
next day, the yardmaster must consider the various costs described above.
He must meet changes in traffic volume with a proper mixture of changes
!o One may wonder why the backlog is not treated explicitly as a queue and why
queueing theory is not used to determine the expected backlog and waiting time. It
would seem more natural, and it might seem to avoid some of the crudeness in our
handling of waitingtime costs and productivity. There are several reasons. One
reason is that the relatively simple queueing models for which analytical solutions are
available do not seem to fit the situation in freight yards very well (2). Monte Carlo
methods would have to be used and it would be difficult to include the results in an
analytical model resulting in simple scheduling rules. Moreover, if the analytical
model were cast aside and if numerical methods were used throughout, it would be
difficult to get any sort of general solution to the scheduling problem we consider.
11 Some of the backlog costs are intangible, difficult to express in dollar terms and
rather crudely handled. For example, waitingtime or delay costs for cars are part of
the backlog costs. In the freight yard studied, a rough estimate by the yardmaster and
other officials of the average relationship between waitingtime costs and backlog was
used. (In simple queueing models, there are explicit relationships between expected
queue length and expected waiting time (6); but they refer to the steadystate and
the models do not seem to be very good aproximations here (2). In view of these
difficulties, it fortunate that moderate errors in the cost parameters have little effect
on the efficiency of the rules. See (3, p. 15).
12 The number of such cars depends on the distribution over time of the train and
car arrivals. In the freight yard studied below, we relied on historical data and the
yardmaster's judgment in determining the average number of cars that arrived so late
that they could not possibly be switched before midnight.
13 These costs are approximated by a function of form: a 6 (Ki — Kii)' 2 .
124 PLANNING AND CONTROLLING PRODUCTION LEVELS
in the number of crews, changes in overtime, and changes in backlog. By
a proper mixture, we mean one that in some sense minimizes costs. The
problem of finding such a mixture is clearly not an easy one.
If the costs can be approximated by quadratic functions, it is possible
to determine linear decision rules to aid the yardmaster in his choice. Such
rules we derived for one large freight yard in the Midwest. Cost data were
gathered there, and the quadratic approximations seemed to provide a
reasonably good fit to the actual cost functions over the normal range.
The quadratic cost function that resulted was:
N
\^i — 2u y*i
i=l
N
= £ 91Ki + ,S5Si  6SKi + .0064 (Si  80/Q 2
+ .004 (B t  400) 2 + (K {  K^ ), 2 Eq. 1.
where d is the total cost for N days, d is the total cost on the /th day, 14
K t is the number of crews used on the /th day, S 4 is the number of cars
switched on the /th day, and Bi is the backlog at the end of the /th day.
There are six terms on the right in Eq. 1. The first term represents the
straighttime crew costs. The next three terms approximate the overtime
costs. The fifth term represents the backlog costs. The last term approxi
mates the costs associated with changes in the number of crews. By de
finition,
Bi = £,_! +A i S i Eq. 2.
where A { is the number of cars that arrive to be switched on the /th day.
Holt, Modigliani, Simon, and Muth (5) (6) have shown that linear
decision rules can be derived that will minimize the expected value of
Eq. 1. under rather general conditions. These rules stipulate how K { and S\
should be chosen. In the freight yard under consideration, these rules are:
Ki = .01091/4, + .00118^f i+1 — .00017^ +2 — .00006/4 i+3
+ .05171^_! + .010915 i _ 1  4.62984 Eq. 3.
Si = .91245/1, + .07937/4, +1 — .01728/4 i+2 — .00487^, +3
+ 2.43193K,_ 1 + .91245B,_!  364.36 Eq. 4.
where A { is the forecasted number of cars that will arrive on the /th day
to be switched.
COST COMPARISONS: RULE VS. ACTUAL
Some evidence concerning the potential usefulness of the rules may be
gathered by comparing the actual cost for some period with the hypothetical
cost had the rules been used. By its very nature, this evidence can only be
14 By total costs, we mean the sum of the straighttime, overtime, backlog, and
crewchange costs.
LINEAR DECISION RULES AND FREIGHT YARD OPERATIONS
125
tentative and suggestive rather than conclusive. Moreover, the evidence
presented here refers only to the one yard where data were gathered.
At this yard, a comparison of the actual and hypothetical performance
was made for a twomonth period in 1955. 15 To make such a comparison,
certain hypothetical or assumed forecasts had % to be used. 10 First, we as
sumed that the yardmaster forecasted traffic perfectly. In this case, the
rule would have smoothed the switching output somewhat, but the general
movement over time would have been similar to the actual movement.
There would have been greater changes from day to day in the number
of crews, but fewer crews would have been employed on the average. The
backlog would have varied less, and it would have hovered about the
optimum level. The rule's performance in terms of cost would have been
better than the actual performance. Average daily costs are shown in Table
I. The overtime and backlog costs seem to be reduced sharply, and the
decrease in straighttime crew costs is also substantial. Total costs are
reduced by about 10 percent, the total saving amounting to about 100,000
dollars a year.
Second, we assumed that the yardmaster could not forecast traffic at all.
We assume that he forecasted for each day the level of traffic that arrived
one week previous to that day. In this case, the movement over time of
the switching would not have been smoothed. Greater changes would have
occurred from day to day in the number of crews, but fewer would have
been employed. The backlog would have varied greatly. 17 Average daily
costs are shown in Table I. Even on the basis of these naive forecasts,
the rule's performance seems slightly superior to actual performance. But
the estimated savings amount to only about .6 percent or $6,500 a year.
TABLE I.
COMPARISON OF ACTUAL COSTS AND
UNDER THE RULES
{Daily costs in dollars)
COSTS
With Rules
Cost Category
Based on
Perfect
Actual Forecast
Based on
Naive
Forecast
Based on
Yardmaster's
Forecast
Straight Time
Overtime
Backlog
Crew Change
$2313 $2232
139 22
72 2
4 7
$2248
33
207
25
$2216
32
208
32
Total
$2528 $2263
$2513
$2488
15 For a discussion of the way in which these hypothetical costs are derived and
their limitations, see (3). Note too that the costs considered here are only the costs
at this one yard. Any secondary effect on other yards is ignored.
10 No records were available concerning past forecasts made by the yardmaster.
17 The interquartile range for the backlog would have been 280 cars. The actual
performance of the yard was such that it equalled 220 cars.
126 PLANNING, AND CONTROLLING PRODUCTION LEVELS
These two comparisons suggest that the indicated savings vary con
siderably with the accuracy of the forecasts. Hence, a third comparison
seems worthwhile. In this comparison, the yardmaster's actual forecasts are
used. For fourteen days, we recorded his forecasts of the number of cars
that would arrive to be switched during each of the next four days. His
forecasting errors were computed, and hypothetical forecasts with the same
error pattern were formulated for the original twomonth period. Had the
rules been used with these forecasts, average daily costs would apparently
have been about 1.5 percent lower than actual costs. Although this saving
may appear to be modest, it would amount to about $15,000 a year
at this yard alone. If such a saving were realized at every large yard, the
railroad we studied would gain well over $500,000 a year. (The relevant
cost comparison appears in Table I.)
Finally, the costs with the rule and perfect forecasts differ greatlv from
the costs with the rule and the yardmaster's forecasts. 18 (The former are
about 9 percent below the latter.) To the extent that this cost differential
is a crude measure of the cost of forecasting errors, it appears that the
installation of equipment and techniques to improve the yardmaster's
forecasts might be worthwhile. As matters stand, his forecasts do not seem
greatly superior to the naive forecasts. (The costs with the rule based on
his forecasts are only 1 percent lower than the costs with the rule based
on the naive forecasts.) The yardmaster could never forecast perfectly,
but the 9 percent differential may be some indication of the maximum
saving from improved forecasting
REACTION TO THE RULES
Though comparisons such as those shown in Table I are some evidence
concerning the potential usefulness of the rules, it was felt that additional
evidence might be obtained by getting the yardmaster's reaction to the
output and employment schedule they prescribe and the problems he en
visages in actually applying them. To gather this type of evidence, an
output and employment schedule was computed from the rule for fourteen
days. 19 Each day, the yardmaster was shown the number of crews and
the switching output that were called for, and he was asked if he could
see any difficulty or disadvantage in the plan. He cited two factors that
in his opinion might constitute problems in using the rules.
18 Note, however, that they are not the yardmaster's actual forecasts during the two
month period. They are based on his forecasts in a later period.
19 Some of the cost coefficients on which this rule was based were not appropriate
to this period, but they seemed to be sufficiently close to the new coefficients that no
adjustment was made. The yardmaster's actual forecasts were used here in deriving
the output and employment schedule.
LINEAR DECISION RULES AND FREIGHT YARD OPERATIONS 127
First, he felt that fluctuations in productivity could be important and
that they might limit the usefulness of the rules. Without detailed data on
productivity changes, it is difficult to assess the importance of this factor.
Productivity changes that can be represented as random variation about a
productivity estimate should cause little difficulty (11, p. 11), and the
changes arising from the heterogeneity of the output and the distribution of
car arrivals can perhaps be represented in this way. 20 But the productivity
changes that represent an appreciable shift in the "productivity probability
distribution" can cause considerable trouble. Hence if there are intersea
sonal differences in productivity, different rules should be used for each
season. 21
To understand the second problem he cited, one should note that the
crew productivity is generally believed to be fairly constant ( all other things
equal 22 ) so long as the number of crews falls in a certain wide interval. If
it falls below this interval, the yard functions poorly; if it falls above this
interval, productivity decreases because of interference. 23 On one day, the
prescribed number of crews fell above this interval, and he cited this as a
second problem. If it occurred frequently, this would indeed constitute a
problem. But judging from the initial twomonth period, it is an extremely
rare occurrence. 24 In the very few instances where it occurs, a simple
remedy may be to monitor the rule. 25
20 The productivity of the crews, like the productivity of service counters in queue
ing theory, is affected by the proportion of time they are idle. This proportion is
clearly a function of the distribution of car arrivals over time. Cars are not homo
geneous with regard to switching. Two groups of twenty cars may be switched, but
because of differences in "cut size" and other factors, one group may take a longer
time to switch than the other. Note that congestioninduced productivity changes are
taken into account in the backlog costs.
21 Interseasonal differences in productivity may occur because of climatic differ
ences and differences in type of traffic. It might also be noted that crews are assigned
to various places in the yard. The rules offer no guidance with respect to crew loca
tion, but the yardmaster seemed to think that this problem was relatively minor.
22 Some of the "things" held constant are distribution of car arrivals, cut size,
climate, degree of congestion in the yard, experience of crews, and amount of cleanup
and other miscellaneous work.
23 Essentially, interference occurs when the functioning of one crew prevents an
other from working effectively.
24 With perfect forecasts, the number of crews never fell outside the interval; with
the yardmaster's forecasts, it fell outside the interval once; with naive forecasts, it
never fell outside the interval.
25 The rule may be monitored by using the number of crews within the interval
that is closest to the number prescribed by the rule. Note too that the rule would al
most surely have to be abandoned during the few instances when a yard is seriously
congested.
128 PLANNING AND CONTROLLING PRODUCTION LEVELS
Finally, the yardmaster was questioned concerning the problem of the
proper planning period. The day is used here as a planning period, but
since some changes in the number of crews can be made during one shift
for the next, each shift could be used as a planning period. Possible ad
vantages in using shifts are that the yardmaster's forecasts may be more
accurate and that the problem of allocating crews and overtime among
shifts is met. His replies shed only a limited amount of light on this matter.
He seemed to feel that a planning period of one day was satisfactory and
that the problem of allocation among shifts was of a secondary order.
CONCLUSION
An attempt has been made here to explore the usefulness of linear de
cision rules in scheduling output and employment in freight yards. The
results refer almost exclusively to one large freight yard where a relatively
intensive investigation was conducted. A comparison of the actual per
formance of the yard with the performance the rules would have prescribed
was fairly encouraging. With forecasts like the yardmaster's, a substantial
saving was shown but it was small percentagewise. The yardmaster seemed
very interested in the rules. He cited two problems; in both cases, there
was a good chance that they would be minor.
On the whole, these results seem fairly encouraging. But note three
things. First, some factors that may be quite important received little at
tention in the model. For example, switching takes place in a time dimension
that is only partially taken into account. Some cars may have to be switched
by a certain time to make proper connections, and the switching rate may
have to hit peaks during the day. Second, some aspects of the model may
not be entirely satisfactory. For example, the treatment of switching output
as strictly a decision variable and the handling of productivity and cardelay
costs seem crude at best. Third, the estimates of the cost coefficients are
sometimes subject to considerable error.
Of course, in any simple model, some factors must be given limited
treatment and some aspects of the formulation seem crude. Discussions
with railroad officials seem to indicate that our treatment of the problem
is reasonably satisfactory to the extent they can judge. In the last analysis,
the test must be the performance of the rules under operating conditions.
Until such a test is conducted, no really informed judgment can be made.
What is needed at this point is a combination of such tests with a refinement
of the model at those places that seem most important in the light of the
tests. The refinement that would be required would probably not be too
difficult; the difficult thing now is to know if, and where, the refinement
should occur.
LINEAR DECISION RULES AND FREIGHT YARD OPERATIONS 129
To conclude, we feel that linear decision rules might be used effectively
in freight yards and that they might in this way contribute to the solution
of an important railroad problem. But until some operating tests are con
ducted, no final judgment can be made.
References
(1) Beckmann, M., McGuire, C, and Winston, C, Studies in Economics
of Transportation, Yale University Press, New Haven, 1956.
(2) Crane, R., Brown, F., and Blanchard, R., "An Analysis of a Railroad
Classification Yard," Operations Research, August 1955.
(3) Holt, C, Modigliani, F., and Simon, H., "A Linear Decision Rule for
Production and Employment Scheduling," Management Science, Octo
ber 1955.
(4) Holt, C, Modigliani, G., and Muth, J., "Derivation of a Linear De
cision Rule for Production and Employment," Management Science,
January 1956.
(5) Holt, C, and Simon, H., "Optimal Decision Rules for Production and
Inventory Control," Proceedings of the Conference on Operations Re
search in Production and Inventory Control, Case Institute of Tech
nology, January 1954.
(6) Kendall, D. G., "Some Problems in the Theory of Queues," Journal of
the Royal Statistical Society, XIII (1951).
(7) Mansfield, E., and Wein, H., "Notes on Railroad Productivity and
Efficiency Measures," Land Economics (forthcoming).
(8) Mansfield, E., and Wein, H., "A Model for the Location of a Railroad
Classification Yard, Management Science (forthcoming).
(9) Mansfield, E., and Wein, H., "A Regression Control Chart for Costs,"
Applied Statistics (forthcoming).
(10) Mansfield, E., and Wein, H., "A Study of DecisionMaking within the
Firm" (mimeographed).
(11) Muth, J. F., "Master Scheduling in a FactoryWarehouse System," ONR
Memorandum 40, Carnegie Institute of Technology.
(12) Simon, H. A., "Dynamic Programming under Uncertainty with a Quad
ratic Criterion Function," Econometrica, January 1956.
Chapter 6
THE PERT SCHEDULING
TECHNIQUE
vm.
Program Evaluation and
Review Technique*
David G. Boulanger
Since the first management "principles" were introduced, most planning
and control methods have been predicated on using historical data. Early
shop practitioners sought the most efficient utilization of time by employ
ing timestudy and tasksetting methods based on stopwatch measurement
of "past" processes that were physical and finite in character. Few useful
techniques have been offered facilitating forward planning of manage
ment activities for which empirical information was not available.
Current industrial activities, however, can be summarized as heavily
oriented toward research and development. A "one best way" of planning
and pursuing R&D projects in terms of most efficient use of time presents
some intangibles that cannot conveniently be measured. This growing con
dition, particularly in defense industries, has prompted the development
of a prognostic management planning control method called Program
Evaluation and Review Technique, or PERT. 1 This article intends to briefly
* From Advanced Management, Vol. 26, No. 4 (1961) 812. Reprinted by permis
sion of Advanced Management.
1 Various acronyms are coined (e.g., PEP, PET, etc.) to describe modifications
from the PERT method described here.
130
PROGRAM EVALUATION AND REVIEW TECHNIQUE
131
present the idea of the PERT method in a manner permitting the reader to
ascertain its potential usefulness. A bibliography provides source material
containing more detailed intricacies of PERT.
The PERT technique was developed as a method of planning and con
trolling the complex Polaris Fleet Ballistic Missile Program for Special
Projects Office (SP) of Bureau of Ordnance, U.S. Navy. The team consisted
of members from SP, the contractor organization, and Booz, Allen and
Hamilton, Chicago. 2 Overall, PERT appears to be a manifestation of the
program concept of management with emphasis on "management by excep
tion," in that potentially troublesome areas in R & D programs can be
spotted and action taken to prevent their occurrence.
PERT, as a dynamic program tool, uses linear programming and statis
tical probability concepts to plan and control series and parallel tasks
which appear only remotely interrelated. Many tasks involve extensive
research and development which itself is difficult to schedule, least of all
to find a "one best way" of doing it. PERT's objective is to determine the
optimum way by which to maximize the attainment, in time, of some pre
determined objective that is preceded by a number of constraints — hence
its linear programming feature. A measure of the degree of risk is predicted
in probabilistic terms to foretell the reasonableness of accomplishment on
scheduled time — hence its statistical probability feature.
PROGRAM NETWORK DEVELOPMENT
The bar chart, presumably derived from Gantt and still widely used,
serves to plan the occurrence of entire phases of tasks in series and parallel
groups over a time period.
An outgrowth of the simple bar chart, called a "milestone chart," in
dicates significant event accomplishments as illustrated in Figure 1.
PROGRAM MILESTONE CHART
TASK A
TASKB
TASKC
TASK D
1 
5
 EVENTS
1
1 I
•
M 1
1 '
'
;m:::;;:
ii
!
«
•
i
TIME'
Figure 1
2 See bibliography Item 5.
132
PLANNING AND CONTROLLING PRODUCTION LEVELS
Neither technique ties together interdependencies between tasks and
significant events. Series and parallel paths should indicate the interrela
tionship constraints between events and tasks as shown by the arrows in
Figure 2.
TIME
PERT NETWORK
V 5
4 >s
TA^ A
1
j —
\ ' 5 \ 2
V \
TASK B
2
7 * si 
<*r
TASKC
3
11
11
j
4 _,
IAoK. U
4 ** 5
Figure 2
A network event describes a milestone, or checkpoint. An event does
not symbolize the performance of work, but represents the point in time
in which the event is accomplished. Each event is numbered for identi
fication.
Arrows connecting events are activities, and represent performance of
work necessary to accomplish an event. No event is considered accom
plished until all work represented by arrows leading to it has been com
pleted. Further, no work can commence on a succeeding event until the
preceding event is completed.
If we include in Figure 2 the estimated weeks to accomplish each
3
activity, e.g., >, the earliest time objective Event 11 above can
be accomplished is the sum of the longest path leading to it. This is the
critical path, and is identified by the heavy lines connecting Events 2, 6,
10, and 11 totaling 17 weeks. The critical path contains the most significant
and limiting events retarding program completion in less than 17 weeks.
But the time required to complete a future task is more realistically stated
in terms of a likelihood rather than a positive assurance. To apply this
likelihood in a probabilistic sense, three time estimates are stated as a
future range of time in which an activity may be accomplished. The three
time estimates are called optimistic, most likely, and pessimistic. They serve
as points on a distribution curve whose mode is the most likely, and the
extremes (optimistic and pessimistic) whose spread corresponds to the
probability distribution of time involved to perform the activity. It is as
sumed there would be relatively little chance (e.g., 1 out of 100) the
PROGRAM EVALUATION AND REVIEW TECHNIQUE
133
activity would be accomplished outside the optimistic or pessimistic time
estimate range. Figure 3 illustrates the estimating time distribution for
completing an activity some time in the future.
FINISH OF EVENT
START ..OPTIMISTIC
PESSIMISTIC
(4 WEEKS)
(12 WEEKS)
Figure 3
From the three time estimates (a, m, b above), a statistical elapsed
a + 4(m)+b c ._^_,__ 3
time (t e ) can be derived by solving t e =
for each activity.*
Following this, a statistical variance O 2 ) can be derived by solving a 2 =
( — ^— ] 2 for each activity. 4 Variance may be descriptive of uncertainty
associated with the three time estimate interval. A large variance implies
greater uncertainty in an event's accomplishment and vice versa, depend
ing whether the optimistic and pessimistic estimates are wide or close
together. This facilitates evaluating risks in a program network, and
using tradeoffs in time and resources to minimize risk and maximize more
efficient use of "factors of production."
PROGRAM NETWORK ANALYSES
The analytical value derived from any PERT network depends on the
configuration and content of the network. Every network should contain
events which, to the program team's best knowledge, serve to significantly
constrain the achievement of the end objective event. Next, events are
3 The elapsed time formula is based on the assumption that the probability density
of the beta distribution f{t) = K(ta) a (bt)v is an adequate model of the distribution
of an activity time.
4 The statistical variance formula assumes the standard deviation as ^(ba).
134
PLANNING AND CONTROLLING PRODUCTION LEVELS
interconnected with "activities" to illustrate their flow and interdepend
encies. After the network of events and activities is defined, three time
estimates for each activity are made.
To illustrate network development and analyses, a hypothetical R&D
program is assumed specifying contract completion 11 months (47 weeks)
after order. Fixed resources are allocated to the program: e.g., 40hour
work week, given personnel, budgeted money, etc. Management now is
interested in:
1 . What's the one best way of conducting effort toward completion?
2. What's the earliest expected time we can complete the program?
3. What are our chances of completing within the contract limitations of
47 weeks?
The network in Figure 4 is a simplified analogue of our plan to develop
a "vehicle armament system." Events are described with a verb in the
past tense to indicate their end accomplishment at a fixed point in time.
PERT NETWORK
VEHICLE ARMAMENT SYSTEM
Figure 4
PROGRAM EVALUATION AND REVIEW TECHNIQUE 135
PERT
ANALYSIS
A
B
C
D
E
F
G
H I
/
K
Event
Pre. Ev.
f«
a 2
Te
Tl
T L T E
T s Pn
T L T E
Event
12
11
5.0
2.78
49.5
49.5
0.0
47.0 .28
0.0
2
5
5.8
.69
0.0
4
9
2.2
.25
0.0
8
10
5.3
1.78
0.0
11
11
8
3
7.8
15.5
4.70
2.25
44.5
44.5
0.0
0.0
12
8
4
17.0
5.44
36.7
36.7
0.0
9.5
7
4
2
5.2
.69
19.7
19.7
0.0
9.5
10
5
2
14.3
2.78
28.8
43.7
14.9
14.9
5
9
6
10.2
3.36
28.5
47.3
18.8
18.8
6
6
2
3.8
.25
18.3
37.1
18.8
18.8
9
10
7
16.0
4.00
34.7
44.2
9.5
20.5
3
7
2
4.2
.25
18.7
28.2
9.5
3
1
8.5
2.25
8.5
29.0
20.5
2
1
_1
14.5
4.70
14.5
14.5
0.0
The analysis of the network is next performed and explained below.
Column A. Each event is listed beginning with objective event back to
the start event.
Column B. The preceding event (s) is (are) listed beside each event.
Hence, there is a succeeding and preceding event for each activity.
Column C. Statistical Elapsed Time (t e ) for each activity is found by sub
stituting optimistic, most likely, pessimistic estimates for a, m, b and
a + 4(m) +b
solving t e — — —  — —  —
6
Column D. Variance (<x 2 ) for each activity is found by substituting
optimistic and pessimistic estimates for a and b and solving o 2 = ( J 2 .
Column E. Earliest Expected Time (T E ) of accomplishment for each
event is found by adding the elapsed time (t e ) of each activity to cumula
tive total elapsed times through the preceding event, staying within a
single path working from "start to finish." When more than one activity
leads to an event, that activity whose elapsed time (t e ) gives the greatest
sum up to that event is chosen as the expected time for that event.
Column F. Latest Time (T L ) for each event is found by first fixing
the earliest time of the objective event as its latest time. Next, the objective
events corresponding elapsed time in Column C (i.e., 5.0 weeks) is sub
tracted to find the latest time of the preceding event, staying within a
single path working backward from finish to start. When more than one
activity leads from an event (while working backwards to determine latest
time), the activity which gives the least sum through that event is selected.
Some events may be accomplished later than expected time and have
no effect on meeting the objective event. Knowledge of "slack" time in
136 PLANNING AND CONTROLLING PRODUCTION LEVELS
a network (i.e., how much and where located) is of interest in determining
program effects of "tradeoffs" in resources from highslack to lowslack
areas.
Since linear programming theory says "negative slack" is not ad
missible (i.e., not technically feasible at the objective event assuming fixed
resources), we commence from an objective viewpoint to compute "posi
tive" (or nonnegative) slack. In spite of theory, a latest time derived from
a fixed contractual date less than earliest expected time must be recognized
to determine how much network "compression" is necessary to meet a
scheduled date with reasonable assurance. The theory simply recognizes
time is not reversible; therefore, to alleviate "negative" slack one must
either extend contractual dates or employ added resources like overtime,
more funds, personnel, etc. We assume in our analyses that resources are
fixed at inception of program to maintain profit potentials.
Column G. Slack time for each event is found by subtracting Expected
Time from Latest Time (T L — T E ). The purpose is 1) to locate the critical
path in the network designated here by events having zero slack and 2) to
determine nextmostcritical paths, as well as those events having substantial
slack.
The critical path contains events most apt to be troublesome technically
or administratively, and are danger points causing potential overall schedule
slippage. Nextmostcritical path(s) is (are) found by substituting next
higher slack event (s) into a second single path from start to finish. For
example, the secondmostcritical path is found by including Events 7 and
10 (whose slack of 9.5 weeks are the nexthigher slack event over the
critical path events) to give a new critical path described as Events 1, 2, 7,
10, 12. Nextmostcritical paths should be observed because their critical
ness may be nearly as severe as the original critical path.
By locating events having substantial slack time, it becomes possible to
effect tradeoffs in resources to those events having little or zero slack. For
example, Events 6 and 9 each have 18.8 weeks slack, meaning their ex
pected time of completion could be intentionally delayed 18.8 weeks with
out causing slippage in overall program schedule. A point of optimization
in network development is approached when the greatest possible number
of events have the smallest possible range in slack from the lowest to highest
slack value.
Column H. Schedule Time (T s ) is the contractual date of completion.
A scheduled time may also exist for major events within a network, which
later facilitates evaluating the range of risks throughout a program plan.
Column I. Probability (P R ) of meeting a scheduled time is calculated to
determine feasibility of program accomplishment under the constraints in
PROGRAM EVALUATION AND REVIEW TECHNIQUE 137
the network. Generally, probability values between .25 and .60 indicate an
acceptable range to proceed with a program as depicted in the network.
Probability values less than .25 assume the schedule time, T s , cannot
reasonably be met with the given resources. Values higher than .60 may
indicate excess resources "built in" the network, and may warrant consid
eration for their use elsewhere. Probabilities need not be computed where
schedule time — T s and expected time — T E are equal, as this assumes .5 or
50 percent probability of completing on schedule.
Probability of events are computed as follows :
(1) Solve for each event which has a schedule time (in our example, the
objective event) :
T a T a 47.049.5 2.5
== —.584
a** 2 * VTOI 4.279
(2) Refer answer to Area Under the Normal Curve Table and compute
probability P R .
The value —.584 refers to —.584 standard deviations from the mean
under a normal curve. Referring to a normal curve table, we find its corre
sponding percent of area under the normal curve to be about —.21904.
Thinking of area under the normal curve and probability as synonymous,
we subtract —.21904 from .50000 (the mean of a normal curve) to derive
a probability of .28906, or 28 percent. Explained, there is a 28 percent
chance of meeting the schedule time of 47.0 weeks, and hence may be
"acceptable" to proceed with the program under plans and resources
factored in the network. Any standard of "acceptability" in probability
terms should be flexible according to the importance of a program and the
consequences if schedule time should not be met. Therefore, any proba
bility value attached to a program plan should be viewed and used
cautiously.
Columns J and K. Under Column J is the ascending order of slack, and
under Column K their corresponding event numbers. This brings out the
Critical Path as 0.0 slack events, nextmost critical path(s), and those
events or paths with high slack from which resources and time may be
deployed to events having zero or low slack. This facilitates locating the
"one best way" of reaching the objective event in relation to time.
* Read as "standard deviation of the sum of the variances." In our example, this is
solved by: 1) finding the sum of the variances ( 2 ) for the events in the Critical
Path, that is 4.70 + .69 + 5.44 + 4.70 + 2.78 = 18.31; 2) finding the square root of
18.31 = 4.279. Probability for any event in a network can be computed if a T s and
T E value is known, and by finding that event within a single network path.
138 PLANNING AND CONTROLLING PRODUCTION LEVELS
REEVALUATION AND OPTIMIZATION
A potential value from PERT at inception of an R & D program is the
opportunity it affords to introduce revised constraints into the plan and then
simulate its outcome. If repeated, the optimum network can be sought, its
troublesome areas located, and various tasks set under optimum conditions
before time, cost, and performance were expended. Computer programs
are available to expedite this, but manual methods are economical for net
works up to 200 events depending on complexity of event interrelation
ships. Various schedules and performance reporting formats can be de
veloped from the analyses for team use and management analysis.
Two advantages from PERT are 1 ) the exacting communications it offers
to participants in a program and 2) its use as a planning foundation to sup
port bid proposals. Each participant can see his relative position and under
stand the timing and relationship of his responsibilities to other participants
on the program team. Often the intangibles and assumptions that plague
accurate bid proposals are brought out when supported with a PERT net
work and analysis.
USING PERT FOR RESOURCES PLANNING,
COST ANALYSIS*
Considerable study is reported with PERT applied to resources (man
power and facilities) planning. Introducing a second variable to create a
simultaneous two dimensional model whose objective functions are to be
optimized — while their preceding constraints are being manipulated (at the
same time satisfying various restrictions placed on potential solution) —
will be more difficult to perform, and probably involve more elaborate pro
cedures.
Some work is reported with PERT applied to cost analysis of a program
(presumably assuming a three or Ndimensional model with variables of
time, resources, cost, et al). It appears the object would be something
analogous to predetermining that point on an average total cost curve where
marginal cost intersects marginal revenue — hence, maximization of profits.
Some suggestions have been offered relative to the "assumed" linearity
between time and cost, in the duration of a program, but this needs clarify
ing before concrete methods of planning costs by PERT can be formulated.
* Notes from American Management Association Meeting, Saranac Lake, N.Y.
March 2729, 1961.
PROGRAM EVALUATION AND REVIEW TECHNIQUE 139
Bibliography
Fazar, Willard, "Progress Reporting in the Special Projects Office," Navy
Management Review; IV, No. 4 (Apr. 1959).
Hamlin, Fred, "How PERT Predicts for the Navy," Armed Forces Manage
ment (July 1959).
Klass, Philip J., "PERT/ PEP Management Tool Use Grows," Aviation Week;
Vol. 73, No. 22 (Nov. 28, 1960), pp. 8591.
Klass, Philip J., "PERT Plan Eases Management Problems," Aviation Week;
Vol. 74, No. 15 (Apr. 10, 1961), pp. 8081.
Malcolm, D. G., J. H. Roseboom, C. E. Clark, and W. Fazar, "Application
of a Technique for Research and Development Program Evaluation," Journal
of Operations Research; Vol. 7, No. 5 (Sept.Oct. 1959), pp. 646669.
PERT Data Processing Lesson Plan Handbook for Technicians, Department of
the Navy, Bureau of Naval Weapons, Special Projects Office; GPO Catalog
No. D217.14: P94, Washington: Government Printing Office, 1960.
PERT Instruction Manual and Systems and Procedures for the Program Evalua
tion System. Department of the Navy, Bureau of Naval Weapons, Special
Projects Office; GPO Catalog No. D217.14: P94/2, Washington: Govern
ment Printing Office, 1960.
PERT Summary Report, Phase I, Department of the Navy, Bureau of Naval
Weapons, Special Projects Office; GPO Catalog No. D217.2: P94/958,
Washington: Government Printing Office, 1960.
PERT Summary Report, Phase II, Department of the Navy, Bureau of Naval
Weapons, Special Projects Office; GPO Catalog No. D217.2: P94/9582,
Washington: Government Printing Office, 1961.
Polaris Management, Department of the Navy, Special Projects Office; GPO
Catalog No. D217.2: P75, Washington: Government Printing Office, 1961.
Proceedings of the PERT Coordination Task Group Meeting {1718 March
1960 and 1617 August 1960), Department of the Navy, Bureau of Naval
Weapons, Special Projects Office; GPO Catalog No. D217.2: P94/ 4/960,
Washington: Government Printing Office, 1960.
Summary Minutes for Meeting of Contractor PERT Reporting Personnel (89
June 1960 and 1415 July 1960), Department of the Navy, Bureau of Naval
Weapons, Special Projects Office; GPO Catalog No. D217.2: P94/5/960,
Washington, Government Printing Office, 1960.
Summary Minutes for Meeting of Contractor PERT Reporting Personnel (15—
16 November 1960, held at U.S. Naval Weapons Laboratory, Dahlgren,
Virginia); GPO Catalog No. D217.2: P94/5/9602, Washington: Govern
ment Printing Office, 1961.
140 PLANNING AND CONTROLLING PRODUCTION LEVELS
IX.
How to Plan and Control with PERT*
Robert W. Miller
The last three years have seen the explosive growth of a new family of
planning and control techniques adapted to the Space Age. Much of the
development work has been done in the defense industry, but the construc
tion, chemical, and other industries have played an important part in the
story, too.
In this article we shall consider what is perhaps the best known of all
of the new techniques, Program Evaluation Review Technique. In par
ticular, we shall look at:
PERT's basic requirements, such as the presentation of tasks, events, and
activities on a network in sequential form with time estimates.
Its advantages, including greatly improved control over complex development
and production programs, and the capacity to distill large amounts of data in
brief, orderly fashion.
Its limitations, as in situations where there is little interconnection between
the different activities pursued.
Solutions for certain difficulties, e.g., the problem of relating time needed and
job costs in the planning stage of a project.
Policies that top management might do well to adopt, such as taking steps
to train, experiment with, and put into effect the new controls.
LEADING FEATURES
The new techniques have several distinguishing characteristics:
(1) They give management the ability to plan the best possible use of
resources to achieve a given goal, within overall time and cost limitations.
* From the Harvard Business Review, Vol. 40, No. 2 (1962), 93104. Reprinted
by permission of the Harvard Business Review.
HOW TO PLAN AND CONTROL WITH PERT 141
(2) They enable executives to manage "oneofakind" programs, as
opposed to repetitive production situations. The importance of this kind of
program in the national and world economy has become increasingly clear.
Many observers have noted that the techniques of Frederick W. Taylor and
Henry L. Gantt, introduced during the early part of the century for large
scale production operations, are inapplicable for a major share of the in
dustrial effort of the 1960's — an era aptly characterized by Paul O. Gaddis
as the "Age of Massive Engineering." 1
(3) They help management to handle the uncertainties involved in pro
grams where no standard cost and time data of the TaylorGantt variety
are available.
(4) They utilize what is called "time network analysis" as a basic method
of approach and as a foundation for determining manpower, material, and
capital requirements.
CURRENT EFFORTS AND PROGRESS
A few examples may serve to indicate for top management the current
status of the new techniques :
The Special Projects Office of the U.S. Navy, concerned with performance
trends in the execution of large military development programs, introduced
PERT on its Polaris Weapon Systems in 1958. Since that time, PERT has
spread rapidly throughout the U.S. defense and space industry. Currently,
almost every major government and military agency concerned with Space Age
programs is utilizing the technique, as are large industrial contractors in the
field. Small businesses wishing to participate in national defense programs will
find it increasingly necessary to develop a PERT capability if they wish to be
competitive in this field.
At about the same time the Navy was developing PERT, the DuPont com
pany, concerned with the increasing costs and time required to bring new
products from research to production, initiated a study which resulted in a
similar technique known as CPM (Critical Path Method). The use of the
Critical Path Method has spread quite widely, and is particularly concentrated
in the construction industry.
A very considerable amount of research now is taking place on the "exten
sions" of PERT and CPM timenetwork analysis, into the areas of manpower,
cost, and capital requirements. As an ultimate objective, "tradeoff" relation
ships between time, cost, and product or equipment performance objectives are
1 See "Thinking Ahead: The Age of Massive Engineering," HBR JanuaryFebruary
1961, p. 138.
142 PLANNING AND CONTROLLING PRODUCTION LEVELS
being sought. This research is being sponsored in two ways — directly by the
military and privately by large companies. Anyone familiar with the current
scene will be impressed by the amount of activity taking place in this field.
For example, at least 40 different code names or acronyms representing varia
tions of the new management controls have come to my attention.
Applications of the new techniques, beyond the original engineeringoriented
programs for which they were developed, are increasing every day. The PERT
approach is usefully introduced in such diverse situations as planning the
economy of an underdeveloped nation or establishing the sequence and timing
of actions to effect a complex merger.
WHAT IS PERT?
Now let us turn to PERT in particular. What are its special character
istics and requirements?
The term is presently restricted to the area of time and, as promulgated
by the Navy, has the following basic requirements:
(1) All of the individual tasks to complete a given program must be
visualized in a clear enough manner to be put down in a network, which is
comprised of events and activities. An event represents a specified program
accomplishment at a particular instant in time. An activity represents the
time and resources which are necessary to progress from one event to the
next. Emphasis is placed on defining events and activities with sufficient
precision so that there is no difficulty in monitoring actual accomplishment
as the program proceeds.
(2) Events and activities must be sequenced on the network under a
highly logical set of ground rules which allow the determination of important
critical and subcritical paths. These ground rules include the fact that no
successor event can be considered completed until all of its predecessor
events have been completed, and no "looping" is allowed, i.e., no suc
cessor event can have an activity dependency which leads back to a
predecessor event.
(3) Time estimates are made for each activity of the network on a three
way basis, i.e., optimistic, most likely, and pessimistic elapsedtime figures
are estimated by the person or persons most familiar with the activity in
volved. The three time estimates are required as a gauge of the "measure
of uncertainty" of the activity, and represent full recognition of the prob
abilistic nature of many of the tasks in developmentoriented and non
standard programs. It is important to note, however, that, for the purposes
of computation and reporting, the three time estimates are reduced to a
HOW TO PLAN AND CONTROL WITH PERT 143
single expected time (t e ) and a statistical variance (o 2 ).
(4) Depending on the size and complexity of the network, computer
routines are available to calculate the critical path through it. Computers
can also calculate the amount of slack (viz., extra time available) for all
events and activities not on the critical path. A negative slack condition can
prevail when a calculated end date does not achieve a program date objec
tive which has been established on a prior — and often arbitrary — basis.
TIME ESTIMATES
Interpretation of the concepts of optimistic, most likely, and pessimistic
elapsed times has varied over the past few years. The definitions which, in
my opinion, represent a useful consensus are as follows:
• Optimistic — An estimate of the minimum time an activity will take, a
result which can be obtained only if unusual good luck is experienced and
everything "goes right the first time."
• Most likely — An estimate of the normal time an activity will take, a result
which would occur most often if the activity could be repeated a number of
times under similar circumstances.
• Pessimistic — An estimate of the maximum time an activity will take, a
result which can occur only if unusually bad luck is experienced. It should
reflect the possibility of initial failure and fresh start, but should not be influ
enced by such factors as "catastrophic events" — strikes, fires, power failures,
and so on — unless these hazards are inherent risks in the activity.
The averaging formulas by which the three time estimates are reduced to
a single expected time (t e ), variance {or 2 ) and standard deviation (o) are
shown in Appendix A. The approximations involved in these formulas are
subject to some question, but they have been widely used and seem appro
priate enough in view of the inherent lack of precision of estimating data.
The variance data for an entire network make possible the determination
of the probability of meeting an established schedule date, as shown in
Appendix B.
CRITICAL PATH
In actual practice, the most important results of the calculations involved
in PERT are the determination of the critical path and slack times for the
network. Exhibit I contains data on the critical path and slack times for
a sample problem (they are based on the method of calculation given in
144
PLANNING AND CONTROLLING PRODUCTION LEVELS
EXHIBIT I.
SLACK
ORDER REPORT
PERT SYSTEM
Airborne Computer — Slack Order Report
Date 7/12/61
Week
0.0
Time in Weeks Page 1
Event
T E
T L T,
Te
T s
P r
001
0.0
0.0
Te = Expected event date
010
7.2
7.2
Oil
12.2
12.2
T L = Latest allowable event date
008
14.5
14.5
TlTe = Event slack
009
19.5
19.5
T s = Scheduled event date
013
21.5
21.5
014
23.5
23.5
23.5
.50
P r = Probability of achieving T s
date
020 20.6 21.5 + .9
019 15.6 16.5 + .9
012 14.4 15.3 + .9
018 9.4 10.3 + .9
007
18.2
20.3
+2.1
006
16.0
18.1
+2.1
005
13.2
14.3
+2.1
003 14.2 19.5 +5.3
Appendix C). The data are shown in the form of a slack order report
(lowest to highest slack), which is perhaps one of the most important
output reports of PERT.
Other output reports, such as event order and calendar time order reports,
are also available in the PERT system.
The actual utilization of PERT involves review and action by responsible
managers, generally on a biweekly basis. Because time prediction and per
formance data are available from PERT in a "highly ordered" fashion (such
as the slack order report), managers are given the opportunity to concen
trate on the important critical path activities. The manager must determine
valid means of shortening lead times along the critical path by applying
new resources or additional funds, which are obtained from those activities
that can "afford" it because of their slack condition. Alternatively, he can
reevaluate the sequencing of activities along the critical path. If necessary,
those activities which were formerly connected in a series can be organized
on a parallel or concurrent basis, with the associated tradeoff risks in
volved. As a final, if rarely used, alternative, the manager may choose to
change the scope of work of critical path activities in order to achieve a
given schedule objective.
HOW TO PLAN AND CONTROL WITH PERT 145
It should be pointed out that the PERT system requires constant updating
and reanalysis; that is, the manager must recognize that the outlook for the
completion of activities in a complex program is in a constant state of flux,
and he must be continually concerned with problems of reevaluation and
reprograming. A highly systematized method of handling this aspect of
PERT has been developed. An example of the input transaction document
involved is given in Exhibit II.
BENEFITS GAINED
Perhaps the major advantage of PERT is that the kind of planning re
quired to create a valid network represents a major contribution to the
definition and ultimate successful control of a complex program. It may
surprise some that network development and critical path analysis do, in
fact, reveal interdependencies and problem areas which are either not
obvious or not well defined by conventional planning methods. The creation
of the network is a fairly demanding task, and is a surefire indicator of an
organization's ability to visualize the number, kind, and sequence of activi
ties needed to execute a complex program.
Another advantage of PERT, especially where there is a significant
amount of uncertainty, is the threeway estimate. While introducing a
complicating feature, this characteristic does give recognition to those
realities of life which cause difficulties in most efforts at planning the future.
The threeway estimate should result in a greater degree of honesty and
accuracy in time forecasting; and, as a minimum, it allows the decision
maker a better opportunity to evaluate the degree of uncertainty involved
in a schedule — particularly along the critical path. If he is statistically
sophisticated, he may even wish to examine the standard deviation and
probability of accomplishment data, which were mentioned previously as
features of PERT. (If there is a minimum of uncertainty in the minds of
personnel estimating individual activity times, the singletime approach
may, of course, be used, while retaining all the advantages of network
analysis.)
And, finally, the common language feature of PERT allows a large
amount of data to be presented in a highly ordered fashion. It can be said
that PERT represents the advent of the managementbyexception principle
in an area of planning and control where this principle had not existed with
any real degree of validity. An additional benefit of the common language
feature of PERT is the fact that many individuals in different locations or
organizations can easily determine the specific relationship of their efforts
to the total task requirements of a large program.
146 PLANNING AND CONTROLLING PRODUCTION LEVELS
This particular benefit of PERT can represent a significant gain in the
modern world of largescale undertakings and complex organizational
relationships.
COPING WITH PROBLEMS
A new and important development like PERT naturally is attended by
a certain amount of confusion and doubt. PERT does indeed have its
problems. However, they are not always what businessmen think they are,
and often there is an effective way of coping with the restrictions. In any
event, it is time to compare the situations in which PERT works best with
situations in which real (or imagined) troubles occur.
UNCERTAIN ESTIMATES
One key question concerns the unknowns of time and resources that
management frequently must contend with.
In PERT methodology an available set of resources including manpower
and facilities is either known or must be assumed when making the time
estimates. For example, it is good practice to make special notations
directly on the network when some special condition (e.g., a 48hour rather
than a 40hour week) is assumed. Experience has shown that when
a wellthoughtthrough network is developed in sufficient detail, the first
activity time estimates made are as accurate as any, and these should not
be changed unless a new application of resources or a tradeoff in goals is
specifically determined. A further caution is that the first time estimates
should not be biased by some arbitrarily established schedule objective, or
by the assumption that a particular activity does not appear to be on a
critical path. Schedule biasing of this kind, while it obviously cannot be
prevented, clearly atrophies some of the main benefits of the technique —
although it is more quickly "discovered" with PERT than with any other
method.
Because of the necessity for assumptions on manpower and resources,
it is easiest to apply PERT in projectstructured organizations, where the
level of resources and available facilities are known to the estimator. PERT
does not itself explicitly resolve the problem of multiprogram planning and
control. But there is general recognition of this problem, and considerable
effort is being devoted to a more complete approach to it. Meanwhile, in
the case of common resource centers, it is generally necessary to undertake
HOW TO PLAN AND CONTROL WITH PERT
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148 PLANNING AND CONTROLLING PRODUCTION LEVELS
a loading analysis, making priority assumptions and using the resulting data
on either a threetime or singletime basis for those portions of the network
which are affected. It should be pointed out, however, that in terms of
actual experience with PERT, the process of network development forces
more problems of resource constraint or loading analysis into the open for
resolution than do other planning methods.
Although PERT has been characterized as a new management control
approach for R & D effort, it has perhaps been most usefully applied in
those situations where there is a great deal of interconnection between the
activities of a network, or where there are interface connections between
different networks. Certainly, network development and critical path analy
sis are not too appropriate for the pure research project, where the capa
bilities of small numbers of individuals with highly specialized talents are
being utilized at a "constant rate" and where their activities have no sig
nificant dependence on other elements of the organization.
JUSTIFYING THE COST
One of the most frequently raised objections to PERT is the cost of its
implementation. A fundamental point to examine here is whether or not a
currently established planning system is giving value commensurate with its
cost — or perhaps more basic still, whether the system is used at all effec
tively to pinpoint and control problem areas. It is quite true that, by the
very nature of its logical requirements for networking, the PERT approach
calls for a higher degree of planning skill and a greater amount of detail
than is the case with conventional methods. In addition, the degree of detail
— or the "level of indenture," as it is called — is a function of:
1 . What is meaningful to the person or persons who will actually execute the
work.
2. The depth of analysis that is required to determine the valid critical path
or paths.
It is perhaps more appropriate to view the implementation of PERT as
costing initially something in the order of twice that of a conventional
planning system. This figure will vary significantly with such factors as:
The degree of planning capability already available.
The present effectiveness and homogeneity of the organization.
The amount and quality of PERT indoctrination given.
The advocates of PERT are quick to point out that the savings achieved
through better utilization of resources far outweigh the system's initial
implementation costs. This better utilization of resources is achieved
through concentration on critical path activities — for example, limiting
HOW TO PLAN AND CONTROL WITH PERT 149
overtime effort to these key activities as opposed to acrosstheboard use of
overtime. Even more important are the "downstream" savings which are
achieved by earlier and more positive action on the part of management to
resolve critical problems.
USE OF STANDARD NETWORKS
Because of the considerable impact of PERT on many organizations
where detailed planning has not had major emphasis, a trend has recently
developed which can be characterized as "model or standard networking."
This has to do with efforts to use the typical or established pattern of carry
ing out a new program in a particular industry. Model networking has many
advantages (particularly in handling the large amounts of data involved in
PERT), but it may also compromise one of the real objectives of PERT —
i.e., obtaining a valid network which is meaningful to the person or persons
who will actually execute the work. In the area in which PERT is used most
effectively, no two programs are ever exactly the same, and no two indi
viduals will have exactly the same approach to the development of a net
work. Therefore, model networks should be introduced with this caution:
management should always allow for the possibility of modifications which
will match the realities of the program.
In addition, the introduction of socalled "master plan networks" and
the topdown structuring of networks for large programs involving many
different firms, while very necessary from the point of view of longrange
planning and the ultimate management of such programs, should be handled
with a philosophy of flexibility. The cardinal principle is that a management
control structure is no better than the adequacy and accuracy of the data
at its base. In the future, the topdown structuring approach — which is
already evident on some major defense and space programs — will probably
increase; but internal objectives, at least, will be subject to reconfirmation
or realignment at the level of industry, depending upon the development of
actual operating networks. The topdown structuring approach is necessary,
however, in order to preserve the mechanics of network integration; it is
important that the data from lower level networks be properly and mean
ingfully summarized into higher level management data.
APPLICATION TO PRODUCTION
A final problem, and one that is often viewed as a disadvantage of the
PERT technique, is the system's lack of applicability to all of the manu
facturing effort. As has been stated, PERT deals in the time domain only
and does not contain the quantity information required, by most manu
facturing operations. Nevertheless, PERT can be, and has been, used very
150 PLANNING AND CONTROLLING PRODUCTION LEVELS
effectively through the preliminary manufacturing phases of production
prototype or pilot model construction, and in the assembly and test of final
production equipments which are still "high on the learning curve." After
these phases, established production control techniques which bring in the
quantity factor are generally more applicable.
Note, however, that many programs of the Space Age never leave the
preliminary manufacturing stage, or at least never enter into mass produc
tion. Therefore, a considerable effort is going forward at this time to
integrate the techniques of PERT within some of the established methods
of production control, such as lineofbalance or similar techniques that
bring in the quantity factor.
COMPUTER OR NO COMPUTER
As a result of the Navy's successful application of PERT on the Polaris
program, and other similar applications, there is a common impression that
the technique is only applicable when largescale dataprocessing equipment
is available. This is certainly true for large networks, or aggregations of
networks, where critical path and slack computations are involved for
several hundred or more events. It is as desirable to have a computer
handle a PERT problem when a large volume of data is involved as it is to
use a computer in any extensive dataprocessing assignment.
Probably equally significant is the fact that several ingenious manual
methods have been developed in industry by those organizations which
have become convinced of PERT's usefulness. These manual methods
range from simple inspection on small networks to more organized but
clerically oriented routines for determination of critical path, subcritical
path, and slack times on networks ranging from fifty to several hundred
events.
This is sufficient proof that PERT can be applied successfully to smaller
programs wherever the degree of interconnection and problems of un
certainty warrant it. For those organizations practiced in the technique,
both the creation of small networks and the formation of time estimates and
their reduction to critical path and slack analyses can be done in a matter
of hours. Exhibit I shows the network for a relatively small electronics pro
gram. Developed in less than a day, the whole network required only two
hours for manual computation.
It seems clear that the small business organization which wishes to par
ticipate in national defense and space programs, or to improve its own
internal schedule planning and control, should not hesitate to adopt PERT
merely because it does not possess largescale dataprocessing equipment.
HOW TO PLAN AND CONTROL WITH PERT 151
PERT EXTENSIONS
Variations of PERT to accommodate multiproject and manufacturing
situations have already been mentioned, and these are merely representative
of a basic movement to extend the approach into the areas of manpower,
cost, and the equipment performance variable. The ultimate objective of
these efforts is to quantify the tradeoff relationships which constantly come
up in development programs but are rarely acted on with explicit data in
hand.
Though none of these extensions have as yet attained as much maturity
and acceptance as PERT, anyone familiar with the current scene will be
impressed by the amount of effort being given to them throughout the
country in both the military and industry. One healthy offset to this par
ticular trend is the fact that the U.S. Air Force has withdrawn its code
name PEP (Program Evaluation Procedure), which was an equivalent for
PERT. There remains, however, a great need for government agencies to
standardize input and output requirements of basic PERT time before uni
formly effective extensions can be made into the area of PERT cost.
COST OF PERT
Much of the research effort on the new management controls which has
taken place throughout the country is concentrated on the problem of man
power and cost. This is probably a reflection of certain facts well known to
most managers of complex development programs:
The jobcosting structures generally found in industry on such programs need
a great deal of interpretation to relate actual costs to actual progress. They are
rarely, if ever, related in any explicit manner to the details of the scheduling
plan.
Cost constraints, either in the form of manpower shortages or funding restric
tions, have a great deal to do with the success with which a program of this type
can be managed.
It seems clear that both of these problems must be solved in any valid
PERT cost approach.
SOLUTIONS REQUIRED
The first problem means that an explicit relationship must be established
between the time network and the jobcost structure, either on a onetoone
basis for each network activity, or for a designated chain of activities. As a
minimum, it seems clear that more detailed jobcost structures are required
than are currently in general use, although this requirement should present
152 PLANNING AND CONTROLLING PRODUCTION LEVELS
no serious limitation for organizations which possess modern dataprocess
ing methods and equipment.
With regard to the development of actual cost figures from the time net
work, an estimate of manpower requirements, segregated by classification,
is usually considered the easiest place to start, since these requirements
were presumably known at the time the network was established. In fact,
however, the actual summation of such data often reveals a manpower or
funding restriction problem, and forces a replanning cycle if no alternatives
are available. (The summation may also reveal inefficiencies in personnel
loading which can be removed by proper use of slack activities.)
Two other problems that should be mentioned are:
Handling of nonlabor items — The costs for these items are often aggregated
in a manner quite different from that which would result from analysis of a
time network. For example, there is a tendency to buy common materials on
one purchase order for a number of different prototypes, each one of which
represents a distinct phase of progress in the program. A refined allocation pro
cedure may be needed to handle this problem.
Coordination and control efforts (e.g., those carried out by project or systems
engineering 2 ) — These are often not indicated on time networks unless they
result in specific outputs. For PERT costing, the network in all cases must be
complete, i.e., it must include all effort which is charged to the program. This
is one of the areas of deficiency in many presentday networks, and one which
must be overcome before an effective PERT cost application can be made.
Each of the foregoing problems can be handled if the underlying network
analysis is sound and subject to a minimum of change. As a result, a
number of different approaches are being attempted in the development of
costed networks which have as their objective the association of at least one
cost estimate with a known activity or chain of activities on the network.
The ultimate objective of all this is not only improvement in planning
and control, but also the opportunity to assess possibilities for "trading off"
time and cost, i.e., adding or subtracting from one at the expense of the
other. It is generally assumed that the fundamental relationships between
time and cost are as portrayed in Exhibit ffl. Curve A represents total direct
costs versus time, and the "U" shape of the curve results from the assump
tion that there is an "optimum" timecost point for any activity or job. It is
assumed that total costs will increase with any effort to accelerate or delay
the job away from this point.
Some companies in the construction industry are already using such a
timecost relationship, although in a rather specialized manner:
2 See Clinton J. Chamberlain, "Coming Era in Engineering Management," HBR
SeptemberOctober 1961, p. 87.
HOW TO PLAN AND CONTROL WITH PERT 153
EXHIBIT III. ASSUMED TIMECOST RELATIONSHIPS FOR A JOB
CURVE D  TOTAL COSTS (A+B+C)
FINAL OPTIMUM POINT
CURVE C UTILITY COSTS S
CRASH POINT
\
.•• CURVE BNONDIRECT COSTS
3Ki
LINEAR APPROXIMATION
NORMAL POINT
f CURVE A  DIRECT COSTS
OPTIMUM POINT
TIME
154 PLANNING AND CONTROLLING PRODUCTION LEVELS
In one application, an assumption is made that there is a normal job time
(which might or might not coincide with the theoretical optimum), and that
from this normal time, costs increase linearly to a crash time, as indicated in
Exhibit IV. This crash time represents the maximum acceleration the job can
stand. On the basis of these assumptions, a complete mathematical approach
and computer program have been developed which show how to accelerate
progress on a job as much as possible for the lowest possible cost. The process
involves shortening the critical path or paths by operating on those activities
which have the lowest timecost slopes.
CHALLENGE OF COST DATA
Making timecost data available for each activity in usable form is one
of the fundamental problems in using PERT in development programs. At
the planning stage, in particular, it is often difficult to determine timecost
relationships in an explicit manner, either for individual activities or for
aggregates of activities. (There are often good arguments for characterizing
timecost relationships at this stage as nonlinear, flat, decreasing, or, more
likely, as a range of cost possibilities.) If alternative equipment or program
objectives are added as a variable, the problem is further compounded.
While posing the problem, it should be pointed out that solutions for the
technical handling of such data, in whatever form they are obtained, have
recently been developed.
Curve B of Exhibit III indicates total nondirect costs, which are assumed
to increase linearly with time. Clearly, accounting practices will have to be
reviewed to provide careful (and probably new) segregations of direct
from nondirect costs for use in making valid timecost tradeoff evalua
tions.
Curve C is a representation of a utility cost curve, which is needed to
complete the picture for total timecost optimization (indicated as the final
optimum point on Curve D). The utility cost curve represents a quantifica
tion of the penalty for not accomplishing the job at the earliest possible
time, and is also shown as a linear function increasing with time.
The difficulties of determining such a curve for many programs, either in
terms of its shape or dollar value, should be obvious. But it is significant to
note that in certain industrial applications such utility cost data have al
ready been developed, typically in the form of "outage" costs or lossof
profit opportunities, and used as the basis for improved decision making.
Further, in the military area, utility cost is the converse of the benefit con
cept in the benefitcost ratio of a weapon system; this factor varies with the
time of availability of a weapon system, even though judgments of benefit
are made difficult by rapidly changing circumstances in the external world.
HOW TO PLAN AND CONTROL WITH PERT 155
CONCLUSION
It is clear that there are difficulties yet to be overcome in advancing the
new management controls — particularly in the new areas into which PERT
is being extended. Yet it is equally clear that significant progress has been
made during the last few years ..Assuming that developments continue at
the rate at which they have taken place up to this time, what position should
top management adopt today with regard to its own internal policies on the
new management controls? Here are the most important steps :
( 1 ) Management should review its present planning and scheduling methods
and compare their effectiveness with that of the PERT system. (I refer here to
time networks only — not timeandcost networks.) If the company has no direct
experience with PERT, it will certainly want to consider training and experi
mentation programs to acquaint the organization with the technique. Manage
ment may even decide to install PERT on all of its development programs (as
some companies have done), even though it has no contractual requirement
to do so.
(2) Management may wish to enter directly into research efforts on the
new management controls or, if such efforts are already underway in the organi
zation, place them on a higher priority basis. As a minimum, it will probably
want to assign someone in the organization to follow the numerous develop
ments that are taking place in the field.
(3) Executives should consider carefully the problem of organization to
make the most effective use of the new management controls. They should
consider the responsibilities of the level of management that actually uses PERT
data in its working form, and the responsibilities of the levels of management that
review PERT in its various summary forms. Clearly, the usefulness of the new
management controls is no greater than the ability of management actually to
act on the information revealed. It should be realized that problems of "re
centralization" will probably accompany the advent of the new tools, particu
larly when applied to the planning and control of large projects throughout an
entire organization.
(4) Finally, management may wish to assess the longer range implications
of the new management controls, both for itself and for the entire industrial
community, since the forces calling for centralization of planning and control
within the firm can apply equally well outside it. In the Age of Massive Engi
neering, the new controls will be utilized to an increasing extent in the nation's
defense and space programs, which are in turn increasing in size and com
plexity. It seems clear that the inevitably closer relationships between govern
ment and industry will require the establishment of new guidelines for procure
ment and incentive contracting where these management control techniques are
used.
156
PLANNING AND CONTROLLING PRODUCTION LEVELS
APPENDIXES
Readers interested in applying PERT may find it helpful to have a more
precise formulation of certain calculations mentioned earlier in this article.
The mathematics involved is basically simple, as the following material
demonstrates.
Appendix A
EXPECTED TIME ESTIMATE
In analyzing the three time estimates, it is clear that the optimistic and the pes
simistic time should occur least often, and that the most likely time should occur most
often. Thus, it is assumed that the most likely time represents the peak or modal
value of a probability distribution; however, it can move between the two extremes.
These characteristics are best described by the Beta distribution, which is shown in
two different conditions in the figures that follow.
M
where: a = optimistic time
m = most likely time
b = pessimistic time .
M = midrange
t e — expected time
m
HOW TO PLAN AND CONTROL WITH PERT
157
As a result of analyzing the characteristics of the Beta distribution, the final ap
proximations to expected time (t e ), variance (a 2 ), and standard deviation (<r) were
written as follows for a given activity:
1
1. t e — ~ (2m + M)
3
2. o 2
m
1 L a + b \
a + Am + b
3. xr =
ba
The first equation indicates that t e should be interpreted as the weighted mean of
m (most likely) and M (midrange) estimates, with weights of 2 and 1, respectively.
In other words, t e is located one third of the way from the modal to the midrange
values, and represents the 50% probability point of the distribution, i.e., it divides the
area under the curve into two equal portions.
Appendix B
PROBABILITY OF MEETING SCHEDULE TIMES
On the basis of the Central Limit Theorem, one can conclude that the probability
distribution of times for accomplishing a job consisting of a number of activities may
be approximated by the normal distribution, and that this approximation approaches
exactness as the number of activities becomes great (for example, more than 10 ac
tivities along a given path). Thus, we may define a curve which represents the proba
bility of a meeting on established scheduleend date, T s :
• sssSwj?:
P <T S >
.:f$H
Illp&
IBta
ill
•»»iisr: : S
Mm,
lit
111
where:
0*(T*)= 2o 2 (/ ei ) + <T 2 (te 2 ) + . . . 6*{t. n )
T Sl = Scheduled Time (earlier than T E )
T s , = Scheduled Time (later than T E )
* Note: The Beta distribution is analyzed in the PERT Summary Report, Phase I
(Special Projects Office, Department of the Navy, Washington, D.C., July 1958).
158 PLANNING AND CONTROLLING PRODUCTION LEVELS
The probability of meeting the T s date when given T E and a 2 for a chain of activi
ties is defined as the ratio of (1) the area under the curve to the left of T s to (2) the
area under the entire curve. The difference between T s and T E , expressed in units of
a, is:
T 8 Te
This will yield a value for the probability of accomplishing Ts by use of the normal
probability distribution table. Thus:
T 8\ — T E
= — \.2a, Pr (accomplishment of T 8l ) = .12
a
T s .  T*
+ 1.2cr, Pr (accomplishment of Ts 2 ) = .88
a
Appendix C
DETERMINING CRITICAL PATH AND SLACK TIMES
The computation steps required to determine the critical path and slack times for
the network shown in Exhibit I are as follows:
Step 1. Determine t e for every activity on the network in accordance with the
equation:
a + Am + b
t e =
Step 2. Starting with Event No. 001, determine T E (or cumulative T E ) for all suc
ceeding events by summing small t e 's for each activity leading up to the event, but
choosing the largest value for the final T E figure in those cases where there is more
than one activity leading into an event.
Step 3. The process covered in Step 2 is now reversed. Starting with the final
event, we determine the latest allowable time,TL, for each event so as not to affect
critical path event times. For example, Event No. 007, with a T E of 18.2 weeks, can
be delayed up to aT L of 20.3 weeks, before it will affect critical path Event No. 013.
Step 4. The difference between T L and T E , known as slack, is next computed for
each event. These computations are shown in Exhibit I in the form of a slack order
report, i.e., in order of lowest to highest values of positive slack. Note that along the
critical path there is zero slack at every event, since by definition there is no possi
bility of slippage along the critical path without affecting the final event date. In this
example, if the end schedule date of Event No. 014 were set at 23.0 weeks rather than
at 23.5 weeks, there would be 0.5 weeks of negative slack indicated for every event
along the critical path.
Step 5. The computation of variance and of standard deviation for this network is
optional and involves adding the variances for each activity along the critical path,
which are obtained from the formula:
ffi
The interested reader may verify that the variance for final Event No. 014, with a T B
of 23.5 weeks, is 1.46 weeks.
Chapter 7
HEURISTICS IN PRODUCTION
SCHEDULING
A Heuristic Method of
Assembly Line Balancing*
Maurice D. Kilbridge and Leon Wester
The principle of the division of labor when applied to the mass assembly of
manufactured items takes the form of the progressive assembly line. The
work is divided into individual tasks and assigned to consecutive operators
on the line. As the product moves down the line each operator adds to it
his share of the work. The process of apportioning the assembly work
among the operators is known as "line balancing."
Although progressive assembly has been practiced for fifty years 1 in
American industry, some of the basic problems associated with it have not
received adequate attention. Among these is the line balancing problem
which, until recently, was not formulated analytically. Trial and error and
enumeration techniques have been relied upon, with the result that industry
wastes an estimated four to ten percent of operators' time on assembly
lines through unequal work assignments.
* From The Journal of Industrial Engineering, Vol. 12, No. 4 (1961), 29298. Re
printed by permission of The Journal of Industrial Engineering.
1 The first progressive assembly line was started at the Ford Highland Park Plant
in 1913, and Henry Ford is properly credited with its invention. He combined the
long known principles of the division of labor, the fabrication of interchangeable
parts, and the movement of product past fixed work stations into the concept of as
sembly as a continuous process.
159
160 PLANNING AND CONTROLLING PRODUCTION LEVELS
In the past five years several analytical studies of line balancing have
appeared in the literature (1), (2), (3), (4), (5), (6), (7), (8), (9).
The systems proposed in these studies range from rigorous mathematical
techniques to approximation routines written for digital computers. Those
presenting computer programs are based on arbitrarily zoned assembly
lines. Although all the proposed methods have merits, none has the ad
vantage of simplicity. Either the method requires a computer, or only a
person of considerable mathematical competence can cope with it.
The problem is approached here with emphasis on the applicability of
the method to existing conditions in manufacturing plants where computers
and mathematicians are not always available. An algorithmic solution is
desirable, but, as frequently happens in combinatorial problems, an algo
rithm becomes intractable as the problem size increases. Therefore a
heuristic method of line balancing was devised which requires logical
analysis of problem data. It is presented here with emphasis on procedure
rather than on computational details.
ASSEMBLY LINE TERMINOLOGY
A work station is an assigned location where a given amount of work is
performed. Assembly line work stations are generally manned by one
operator. However, on short runs an operator may man more than one
station, and on lines of large products (aircraft, for example) work stations
are frequently manned by several operators.
A minimum rational work element is an indivisible element of work or
natural minimum unit beyond which assembly work cannot be divided
rationally. For example, a minimum rational element may include the fol
lowing motion pattern: reach to a tool, grasp it, move it into position, per
form a single task, return the tool. In practice such work elements are
considered indivisible since they cannot be split between two operators
without creating unnecessary work in the form of extra handling.
The total work content is the aggregate amount of work of the total
assembly. The total work content time is the time required to perform the
total work content. The station work content time is the time required to
perform the work content of the given station. This time is also known as
the operation time.
The cycle time is the time the product spends at each work station on the
line when the line is moving at standard pace, or 100 percent efficiency.
Stated in other words, the cycle time is the amount of time elapsing be
tween successive units as they move down the line at standard pace. Extend
ing this definition, the cycle time is the maximum operation time. The
cycle time, and the pace at which the line operates, together determine the
rate at which products flow from the line.
A HEURISTIC METHOD OF ASSEMBLY LINE BALANCING 161
Balance delay time is the amount of idle time on the line due to the im
perfect division of work between stations. Since it seems to be seldom
possible to divide the work evenly between all operators on the line, those
operators having shorter assignments will have some idle time. This idle
time is a measure of the imbalance of the line.
In practice those operators having shorter work assignments will not
actually stand idle at the end of each cycle, but will work continuously at a
slower pace. The effect, measured in terms of labor cost, however, is the
same as if they were idle part of the time and working at a faster pace the
rest of the time.
If all of the station work content times were equal, there would be no
imbalance. The degree or percent of imbalance, simply called "balance
delay," is the ratio between the average idle time at the stations and the
maximum operation time. Stated otherwise, balance delay is the ratio be
tween the total idle time and the total time spent by the product in moving
from the beginning to the end of the line.
Balancing restrictions are constraints imposed on the order or time
sequence in which work elements can be performed. They are of three
types :
1. Technological restrictions on the order of assembly of components or
piece parts.
2. Restrictions imposed by fixed facilities or machines on the line.
3. Restrictions of position, where position refers to the operator or operators.
THEORETICAL ANALYSIS OF BALANCE DELAY
Balance delay in percent is defined in mathematical terms as the ratio
d = 100 C ^
c
where c is the maximum operation time or cycle time, and c is the average
operation time. If the assembly line is manned by n operators (one at each
of n work stations), then d can also be written as
d = 100 nc ~ n6 = 100 nc ^\ ( nc  % U S3* 0) ,
nc nc
where U (i= 1, 2, ...,&), is the duration of the ith work element in the
distribution and %U is the total work content time.
It is important to note that for a given cycle time, c, and a given total
work content time, %t it there exists a minimum number of operators, w min .
The minimum balance delay in percent is defined as
162 PLANNING AND CONTROLLING PRODUCTION LEVELS
d min = 100 " minC = %ti
and is a function of cycle time, c. This function, called the balance delay
function, can be plotted on a graph. Since the definition of d min is unequivo
cal, the subscript can be dropped, and
^=100 ^^
nc
will have the connotation of d min .
The range of possible cycle times, c, is
'max C Aiti
That is, the cycle time, c, must equal or exceed the maximum element,
/ max , in the work element distribution, but cannot exceed the total work
content time, %U. The lower bound of c follows from the definition of cycle
time. The upper bound is based on practical considerations. There is no
sense in allowing the cycle time to increase beyond the total work content
time. For computational convenience it will be assumed that the durations
of work elements t it are expressed in integers, so that %U is also an integer.
A simple but useful theorem in line balancing theory can now be stated
as follows: A necessary but not sufficient condition for perfect balance (or
zero delay) is that
nc — %iti = 0,
where n (the number of work stations) is an integer. This implies that %t*
must be divisible by c, for otherwise n == %t { /c is not an integer.
The condition is necessary, because if balance delay is zero, then
nc — %ti = 0, and %U/c == n is an integer (by assumption).
It is not sufficient because situations exist for which, although %U/c is an
integer, it is not possible to assign the given elements to the n work stations.
In order to find all possible zero points of the balance delay function,
%U must be divided by all the c's, (/ max ^ c ^%t t ), respectively, for which
the quotient %U/c is an integer, n. Then if, for a given cycle time, c, all the
work elements in the distribution can be fitted into the n work stations or
intervals of duration c each, while heeding sequential restrictions, perfect
balance has been attained for that particular case. Otherwise, the number,
n, of work stations must be increased by the least integer rquired to achieve
balance, and then the balance is not perfect (d > 0). For all other cycle
times, it is known a priori that the balance delay is not zero. To find the
corresponding balance delay value, the number of work stations is chosen
as an integer exceeding %tjc by the minimum amount required to fit the
elements into the n intervals.
A HEURISTIC METHOD OF ASSEMBLY LINE BALANCING
163
Since the work element times are expressed in integers, the cycle times
are integers also, and the balance delay function, d = /(c), is a discrete
function. However, it will be convenient to plot its graph by connecting
consecutive points of the function with straight lines. The graph will thus
have the appearance of continuity and will emphasize features and trends
which might otherwise be overlooked.
In general, the balance delay function thus derived has the shape of a
zigzag line reaching the cycle time axis wherever a zero point occurs.
Figure 1 shows a portion of a balance delay curve for an assembly line.
The curve is drawn for cycle times of from 0.80 minutes to 1.20 minutes.
The figure shows that within this range, zero balance delay is attainable at
80 82 84 86
92 94 96 98 100 102 104 106 1(
CYCLE TIME IN 0.01 MINUTES
NUMBER OF OPERATORS
m ir>
NUMBER OF UNITS PRODUCED DURING 8 HOUR SHIFT
Figure 1 . Typical Balance Delay Function, 80 ^ c
120
a cycle time of 0.99 minute. The cycle time actually in use on this line was
0.94 minute, which, according to Figure 1 would entail a minimum balance
delay of about 1 .25 percent. The balance delay actually experienced on the
line was 5.0 percent.
The balance delay function tells what cycle time to select for a given
distribution of work elements and a given number of operators. Since
balance delay is to be minimized, a cycle time should be chosen which, in
conjunction with the given number of operators, insures the least balance
delay. These points may be found for all values of n at the local minima of
the balance delay function.
164 PLANNING AND CONTROLLING PRODUCTION LEVELS
Another question that can be answered from a glance at the balance
delay function is how many units, N, can be produced most economically
on the assembly line in a given time interval, say, during an eighthour
shift, with n operators, or conversely, what cycle time should be used in
order to produce N units with n operators.
Since there is a onetoone correspondence between the cycle time, c,
and the number of units produced, N, during a given time of duration
T
T (N = — ), a horizontal scale indicative of N can be added to the graph
of the balance delay function. For example, Figure 1 indicates that about
461 units can be produced in eight hours with 15 operators at a cycle time
of 1.04 minutes with a minimum balance delay of 4.75 percent. The figure
also shows that 15 operators can produce in eight hours 485 units at a
cycle time of 0.99 minute with zero balance delay. This implies that produc
tion of 461 units will cost as much as that of 485 units. Thus management
has the two least cost alternatives of choosing a production rate of 485
units per 8 hour shift at a cycle time of 0.99 minute, or of producing 461
units at this same cycle time in about a iy 2 hour shift.
THE LINE BALANCING METHOD
To describe the heuristic method of line balancing an illustration is given
which is not drawn from industry. Other examples are available; however,
they involve technological complexities which have no direct bearing on the
method. The illustration is that first suggested by Salveson (6) of dressing
in the morning, that is, of assembling one's clothes. The example has
essentially the same properties as many industrial jobs, and therefore is not
trivial. The work can be divided into a given number of operators of equal
duration while heeding necessary sequential restrictions. Each operation
consists of one or more irreducible work elements combined in such a way
as to equalize the time required for each operation.
The elements and their time durations are shown in Table I. Thus,
element 1 of duration 9 must precede element 3 of duration 10, and element
7 of duration 13. Element 3 must precede element 5 of duration 17 and
element 7 must precede element 14 of duration 22, and so forth.
The precedence diagram is drawn so that the assembly progresses from
left to right, each element being as far left as possible at the start of the
procedure. Precedence diagrams are constructed as described by Jackson
(4). First, in column I of the diagram are listed all work elements which
A HEURISTIC METHOD OF ASSEMBLY LINE BALANCING 165
TABLE I. WORK ELEMENTS IN "ASSEMBLING CLOTHES'
Element Time
No. Description (inO.Olmin.)
1 . Pick up and put on left sock 9
2. Pick up and put on right sock 9
3. Pick up and put on left shoe 10
4. Pick up and put on right shoe 10
5. Tie left shoe lace 17
6. Tie right shoe lace 17
7. Pick up, put on, and attach left garter 13
8. Pick up, put on, and attach right garter 13
9. Pick up, put on left spat 20
10. Pick up, put on right spat 20
11. Pick up and put on undershorts 10
12. Pick up and put on undershirt 11
13. Tuck in undershirt 6
14. Pick up and put on trousers 22
15. Pick up and put on shirt 11
16. Button 5 buttons 19
17. Tuck in shirt 12
18. Button shirt neck button 4
19. Turn up collar 3
20. Pick up and put on tie, turn down collar 7
21. Tie tie 55
22. Pick up and put on tie clip 14
23. Fold back left cuff — pick up and put on left cuff link 27
24. Fold back right cuff — pick up and put on right cuff link 29
25. Pick up and put on belt 26
26. Buckle belt 6
27. Pick up and put on vest 5
28. Button vest (6 buttons) 24
29. Pick up and place handkerchief 4
30. Pick up and place wallet 5
3 1 . Pick up and place small change 7
32. Pick up and place keys 4
33. Pick up and put on suit jacket 15
34. Button suit jacket 7
35. Pick up and place pocket handkerchief 7
36. Pick up and place fountain pen 9
37. Pick up and place glasses 4
38. Pick up and place glass case 3
39. Pick up and put on wrist watch 5
40. Pick up and put on scarf 4
41. Pick up and put on top coat 21
42. Button top coat 12
43. Pick up and put on hat 6
44. Pick up and put on left glove 5
45. Pick up and put on right glove 5
Total work content (Sif f ) 552
166 PLANNING AND CONTROLLING PRODUCTION LEVELS
need not follow any work elements. Then, in column K (K ^ II) are
entered all those work elements which must follow work elements already
on the diagram. Finally, arrows are drawn from work elements in column
K — 1 to work elements in column K which must follow them. This pro
cedure is repeated, replacing column K — 1 by columns K — 2, . . . , 1,
successively, except that no arrow is drawn from one work element to
another if it is possible to follow arrows already drawn from the first work
element to the second.
The problem of line balancing is to achieve the least possible balance
delay for given conditions. Thus, for a specified distribution of elements and
restrictions on their ordering, and a given cycle time, c, one may be required
to find the minimum number, n, of operators to perform the task; or, for a
given number, n, of operators, one may wish to determine the shortest cycle
time, c. If neither c nor n is given, it is necessary to determine the value, or
values, of c and n for which balance delay is zero. This is the procedure
followed below.
To determine the cycle times for which %t { /c is an integer, it is con
venient to write %ti as a product of prime numbers, i.e.,
%U = 552 = 2 X 2 X 2 X 3 X 23.
Since 55 ^ c ^ 552, it is easily seen that the quotient %U/c is an integer
for
d = 2 X 2 X 2 X 3 X 23 = 552
c 2 = 2 X 2 X 3 X 23 =276
c 3 = 2x 2X2X23 =184
c 4 = 2 X 3 X 23 =138
c 5 = 2 X 2 X 23 =92
c 6 = 3 X 23 =69
Hence, perfect balance can possibly be achieved with
%U 552 1 . . ■ :
«i = — = = l station (trivial case)
ci 552
titi
«2=
c 2
552
~ 276"
= 2 stations
2*
c 3
552
"184"
= 3 stations
A HEURISTIC METHOD OF ASSEMBLY LINE BALANCING 167
%U 552
« 4 = — = = 4 stations
c 4 138
%U 552 ^
n 5 = — = = 6 stations
c 5 92
%U 552
n 6 = — == = 8 stations
c Q 69
The question now is whether perfect balance is actually attainable for the
above six cycle times.
Elements which are mutually independent can be permuted among them
selves in any work sequence without violating restrictions on precedence
relations. Furthermore it can be seen that many elements can be moved
without disturbing the precedence restrictions. The flexibility depends on
the number of restrictions, but in most industrial cases this number is
moderate, affecting the flexibility only slightly.
The above two properties of elements in the precedence diagram —
permutability and lateral transferability — are exploited in the attempt to
achieve optimum balance. Other properties which offer additional oppor
tunities will not be discussed here since in most cases they are of minor
interest.
Next in the line balancing procedure is the construction of a table con
taining detailed information about each element taken from successive
columns of the precedence diagram. Thus column A of Table II shows the
column number, and column B indicates the element identification number.
Column C is reserved for remarks concerning the transferability of elements.
For instance, it is indicated under "Remarks" that element No. 39 can be
moved from column I to columns II, III, . . . , up to column XI. It is also
stated that element No. 4 can be moved from columns II to columns III,
IV, . . . , up to column IX provided elements Nos. 6 and 10 are displaced
accordingly, and so forth. Column D gives the time duration for each
element, column E, the sum of time durations for each column in the
diagram, and column F the cumulative time sums.
Although perfect line balance was achieved for all six values of c (552,
276, 184, 138, 92, 69), the heuristic method will be demonstrated only
forc 3 = 184.
168
PLANNING AND CONTROLLING PRODUCTION LEVELS
TABLE II. TABULAR REPRESENTATION OF PRECEDENCE
DIAGRAM FOR WORK ELEMENTS IN "ASSEMBLING CLOTHES'
(A)
Column
Number
of
Diagram
(B)
Element
Identification
Number
(C)
Remarks
(D) (F)
Element (E) Cumula
Time Sum of tive
Duration Time Time
ti Durations Sums
I
1
2
11
12
39
> II,
XI
9
9
10
11
5
44
44
II
3 (w. 5,9)
7
4 (w. 6, 10)
8
13
37 (w. 43)
* HI,..., IX
> III, ...,IX
» III, ..., XIII
10
13
10
13
6
4
56
100
III
5
6
14
15
43
(w. 9)
(w. 10)
IV, . . . , X
IV, . . . , X
IV, ... , XIV
17
17
22
11
6
73
173
IV
9
10
29
30
31
32
25
16
19
23
24
V,
v,
V,
v,
v,
V,
v +
V,VI
V,VI
XI
XI
XI
XI
XI
XI
20
20
4
5
7
4
26
19
3
27
29
164
337
V
17
20
12
7
19
356
VI
26
27
18
* VII, . . . , IX
6
5
4
15
371
VII
21
33 (w. 35, 36,38)
+VIII
55
15
70
441
VIII
22
35
36
38
* IX, X
» IX
» IX
14
7
9
3
33
474
IX
28
24
24
498
X
34
7
7
505
XI
40
4
4
509
XII
41
21
21
530
XIII
42
12
12
542
XIV
44
45
5
5
10
_552
A HEURISTIC METHOD OF ASSEMBLY LINE BALANCING 169
Example for c 3 = 184.
If the work can be balanced perfectly for c 3 = 1 84, then the number of
stations is
_ XU __ 552 _
" 3 = ~V Z = 184 =
Column F of Table II indicates that if time elements can be arranged in a
proper sequence, the cumulative sum will add to 184 within column IV.
Is it possible to permute the elements in column IV, so that this number
(184) is actually obtained? A cursory examination shows that if elements
Nos. 31 and 32 are moved to the top of column IV, the cumulative sum of
time durations up to and including element No. 32 is exactly equal to 184.
Hence, one assigns work elements Nos. 1, 2, . . . , 39, 3, . . . , 37, 5,
. . . , 43, 31, 32, in that order, to the first station (see Table III).
The work sequence for station one is thus: 1. pick up and put on left
sock; 2. pick up and put on right sock; 11. pick up and put on undershorts;
12. pick up and put on undershirt; 39. pick up and put on wrist watch;
3. pick up and put on left shoe; 7. pick up, put on and attach left garter;
4. pick up and put on right shoe; 8. pick up, put on and attach right garter;
13. tuck in undershirt; 37. pick up and place glasses; 5. tie left shoe lace;
6. tie right shoe lace; 14. pick up and put on trousers; 15. pick up and put
on shirt; 43. pick up and put on hat; 31. pick up and place small change;
32. pick up and place keys. Although this may seem to be a curious dress
ing sequence, it is obviously possible, in that it does not contradict any of
the sequencing restrictions.
Referring to column F of Table III, it is evident that the cumulative sum
of 368 (i.e. 184 + 184) will occur within column VI of the diagram, if the
elements can be ordered suitably. However, it is not possible to simply
permute the elements in that column to arrive at the total of 368, and it is
now necessary to resort to another stratagem. By moving elements in the
diagram from their respective columns to positions in succeeding columns,
the desired result is achieved.
Thus it is convenient to make the following changes:
Move element No. 9 from column IV to column VIII,
Move element No. 10 from column IV to column VIII,
Move element No. 25 from column IV to column VIII,
Move element No. 33 from column VII to column VIII,
Move element No. 26 from column VI to column IX,
Move element No. 35 from column VIII to column IX,
Move element No. 36 from column VIII to column IX,
Move element No. 38 from column VIII to column IX.
170
PLANNING AND CONTROLLING PRODUCTION LEVELS
TABLE III. MODIFIED TABLE II AFTER ASSIGNMENT OF
WORK ELEMENTS TO STATION 1 ONLY (f a = 184)
(A)
Column
Number
of
Diagram
(B)
Element
Identification
Number
(C)
Remarks
(D) (F)
Element (E) Cumula
Time Sum of five
Duration, Time Time
ti Durations Sums
1
2
11
12
39
3
7
4
8
13
37
III
5
6
14
15
43
17
17
22
11
6
IV 31
32
184
184
9
— >
V, . . . , XI
20
10
>
V, . . . , XI
20
29
— >
V, . . . , XI
4
30
»
V, . . . , XI
5
25 (w. 26)
>
v +
26
16
19
19
3
23
»
V,VI
27
24
*•
V,VI
29
24
+ V,VI
29
153
337
V
17
20
12
7
19
356
VI
26
27
18
^ VII, . . .
,IX
6
5
4
15
371
VII
21
33
(w
35,
36
38)
* VIII
55
15
70
441
VIII 22
35
36
38
X 7
9
3
IX 28
X 34
XI 40
XII 41
474
498
505
509
530
542
21
12
XIII 42
12
XIV 44
45
10
552
A HEURISTIC METHOD OF ASSEMBLY LINE BALANCING
171
TABLE IV. MODIFIED TABLE III AFTER ASSIGNMENT OF
WORK ELEMENT TO ALL THREE STATIONS (c 3 = 184)
(A)
Column
Number
of
Diagram
(B)
Element
Identification
Number
(C)
Remarks
(D)
Element
Time
Duration,
tt
(E) (F)
Sum of Cumulative
Time Time
Durations Sums
I
1
9 T
2
9
11
10
12
11
39
5
II
3
10
7
13
4
10
8
13 a
13
6 2
37
4 2
III
IV
5
6
14
15
43
31
32
184
184
IX
XIV
29
30
16
19
23
24
9
10
25
33
28
26
35
36
38
44
45
4
5
19
3
27
29
20
20
26
15
24
6
7
9
3
V
17
20
12
7
VI
27
18
5
4
VII
21
55
VIII
22
14
184
368
X
34
7
XI
40
4
XII
41
21
XIII
42
12
184
552
172 PLANNING AND CONTROLLING PRODUCTION LEVELS
After the modification of Table III, the cumulative sum of time durations
equaling 368 falls into column VIII. Thus, elements Nos. 29, ... , 24, 17,
20, 27, 28, 21, 22, in that order, are assigned to the second station, and
the remaining elements Nos. 9 through 45, as indicated in Table IV, are
assigned to the third station. It is thus demonstrated that perfect balance
can be achieved with a cycle time c 3 = 184.
The heuristic method of line balancing was demonstrated for a cycle time
with promise of potential zero balance delay. This was done because per
fect balancing cases are the most difficult to solve. This is true because it is
easier to pack the work elements into cycle times when it is not required
that the time intervals be completely filled. For all cycle times other than the
six values derived, it is known a priori that perfect balance is unattainable.
To find the resulting balance delay the number of stations, n, is chosen as
an integer exceeding %iU/c by the minimum amount required to pack the
elements into these n stations. If balance can be attained with n being the
smallest integer larger than %tjc, optimality is assured. Otherwise care and
judgment must be exercised before the solution can be labeled "optimum."
GENERAL OBSERVATIONS REGARDING
THE HEURISTIC METHOD
The following generalizations and suggestions may prove helpful in ap
plying the heuristic method.
1 . Permutability within columns is used to facilitate the selection of elements
of the length desired for optimum packing of the work stations. Lateral trans
ferability helps to deploy the work elements along the stations of the assembly
line so they can be used where they best serve the packing solution.
2. Generally the solutions are not unique. Elements assigned to a station,
which belong after the assignment is made in one column of the precedence
diagram, can generally be permuted within the column. This allows the line
supervisor some leeway to alter the sequence of work elements without disturb
ing optimum balance.
3. Long time elements are best disposed of first, if possible. Thus, if there is
a choice between the assignment of an element of duration, say, 20, and the
assignment of two elements of duration, say, 10 each, assign the larger element
first. Small elements are saved for ease of manipulation at the end of the line.
The situation is analogous to that of a paymaster dispensing the week's earn
ings in cash. He will count out the largest bills first. Thus, if the amount to be
paid a worker is $77, the paymaster will give three $20 bills first, then one $10
bill, one $5 bill, and two $1 bills, in that order.
A HEURISTIC METHOD OF ASSEMBLY LINE BALANCING 173
4. When moving elements laterally, the move is best made only as far to the
right as necessary to allow a sufficient choice of elements for the work station
being considered.
CONCLUSIONS
The proposed method of assembly line balancing is not a mere mechanical
procedure, since judgment and intuition must be used to derive a meaning
ful solution. However, the procedure is simple and can be used by produc
tion engineers without difficulty. Optimality is usually assured when the
ordering restrictions on the work elements are of a technological nature
only. In a few industrial cases where the sequential restrictions governing
the elements are stringent, caution must be used in labeling the solution
"optimum." If, in addition to technological restrictions, the constraints are
positional or involve fixed plant facilities, such as overhead cranes, testing
booths or other immovable installations, the work elements must be grouped
into subdistributions. The consideration of these cases is beyond the scope
of this presentation.
The heuristic method is an improvement over the trial and error methods
traditionally used in assembly line balancing. Changes in model or in pro
duction processes can be made without creating difficult rebalancing work.
The heuristic method generally provides a solution which is not unique,
thus permitting the line supervisor to choose that solution which minimizes
balance delay and also satisfies secondary objectives.
The method can be applied in any industry using progressive assembly
techniques. It has been used successfully in the electronics and home ap
pliance industries.
References
(1) Bryton, B., "Balancing of a Continuous Production Line," unpublished
M.S. Thesis, Northwestern University, 1954.
(2) Burgeson, J. W., and Daum, T. E., Production Line Balancing, File Num
ber 10.3.002, International Business Machines Corporation, Akron, Ohio,
1958, 62 pp.
(3) Hoffman, T. R., "Permutations and Precedence Matrices with Automatic
Computer Applications to Industrial Problems," unpublished Ph.D.
Thesis, University of Wisconsin, 1959.
(4) Jackson, J. R., "A Computing Procedure for a Line Balancing Problem,"
Management Science, 3 (1956), pp. 261271.
(5) Mitchell, J., "A Computational Procedure for Balancing Zoned Assembly
Lines," Research Report No. 6948011R3, Westinghouse Research
Laboratories, Pittsburgh, 1957, 11 pp.
174 PLANNING AND CONTROLLING PRODUCTION LEVELS
(6) Salveson, M. E., "The Assembly Lfne Balancing Problem," The Journal
of Industrial Engineering, 3 (1955), pp. 18—25.
(7) Tonge, F. M., "Summary of a Heuristic Line Balancing Procedure," The
RAND Corporation, T1799, September 1959.
(8) Tonge, F. M., "A Heuristic Program of Assembly Line Balancing," The
RAND Corporation, p. 193, May 1960.
(9) Tonge, F. M., "Summary of a Heuristic Line Balancing Procedure," Man
agement Science, October 1960, pp. 2139.
PART II
Inventory Control
Chapter 8
INTRODUCTION TO PART II
Inventory control has been, and no doubt will continue to be, one of the
most important areas for operational analysis. It is important because a very
sizable portion of the assets of any typical manufacturing or distribution
enterprise is tied up in inventories. General Electric's investment in inven
tories was over $655 million in 1960. This figure represented 25 percent of
G.E.'s total assets. With this kind of investment in inventory, which again
is very typical in manufacturing firms, the need for planning good inventory
control systems is apparent.
The first three chapters in this section present an outline of the important
considerations for efficient inventory policy. Originally printed as a series
in the Harvard Business Review, these articles have stimulated consider
able interest in inventory control in both academic and industrial circles.
Chapter 9 begins with a discussion of the functions of inventories, some
problems of inventory management, and the nature of inventory costs. The
remainder of the article is devoted to the important question of determining
the economic lot size.
The concept of an inventory system is developed in Chapter 10. The
tiein between production scheduling and inventory control is discussed and
an excellent case example of "the development of an efficient inventory
control system is presented.
Chapter 11 brings the areas of production planning and control into the
inventory picture. Methods for meeting various demand patterns are dis
cussed and integrated with overall inventory policy. This series does an
excellent job of presenting a comprehensive view of inventory control, and
is a good starting point for students interested in obtaining a background
in this area. x
1 The material presented in this series has been expanded by Mr. Magee and is
now available in book form: John F. Magee, Production Planning and Inventory
Control, McGrawHill Book Co., Inc., 1958.
177
17 ° INVENTORY CONTROL
Quite often inventory problems are difficult to handle simply because of
their size. Companies with large distribution operations involving thousands
of products distributed through several warehouses must often set an ex
tremely large number of inventory levels. Chapter 13 concerns the use of
simulation to determine inventory levels in the warehouses of a distribution
company. This article presents a good example of the newer techniques that
are increasingly being brought to bear on inventory problems.
Chapter 12 could well be included in Part Three on Facilities Planning,
since it deals with a new method for evaluating alternative procedures for
producing a given product. The method is actually a combination of break
even analysis and the economic lot size model, however, and is therefore
more closely related to inventory control strategy.
Although breakeven analysis and the economic lot size model may
appear at first to be a strange combination, either technique, used indi
vidually, may lead to some significant errors. Breakeven analysis does not
account for the possibility of producing over several cycles and neglects in
ventory carrying cost. The economic lot size model, on the other hand,
ignores the possibility of alternative manufacturing methods and accom
panying differences in cost. The combined model developed in Chapter 12
is an interesting application of some of the considerations discussed in
Chapters 9 through 11 and provides potentially useful techniques for
evaluating production alternatives.
Finally, in Chapter 14, statistical methods are applied to problems of
inventory control. Thus, the material in Part Two first surveys the inventory
control area and then develops extensions of modern analytical methods to
largescale inventory systems and to dynamic relationships of lotsize and
production alternatives.
Chapter 9
BASIC FUNCTIONS
XL
Guides to Inventory Policy
I. Functions and Lot Sizes*
John F. Magee
"Why are we always out of stock?" So goes the complaint of great numbers
of businessmen faced with the dilemmas and frustrations of attempting
simultaneously to maintain stable production operations, provide customers
with adequate service, and keep investment in stocks and equipment at
reasonable levels.
But this is only one of the characteristic problems business managers face
in dealing with production planning, scheduling, keeping inventories in
hand, and expediting. Other questions — just as perplexing and baffling
when managers approach them on the basis of intuition and pencil work
alone — are: How often should we reorder, or how should we adjust pro
duction, when sales are uncertain? What capacity levels should we set for
jobshop operations? How do we plan production and procurement for
seasonal sales? And so on, and so on.
In this series of articles, I will describe some of the technical develop
ments which aim at giving the business manager better control over inven
tory and scheduling policy. While these techniques sometimes employ con
* From the Harvard Business Review, Vol. 34, No. 1 (1956), 4960. Reprinted by
permission of the Harvard Business Review.
179
180 INVENTORY CONTROL
cepts and language foreign to the line executive, they are far from being
either academic exercises or mere clerical devices. They are designed to
help the business manager make better policy decisions and get his people
to follow policy more closely.
As such, these techniques are worth some time and thought, commen
surate with the central importance of production planning and inventory
policy in business operations. Indeed, many companies have found that
analysis of the functions of inventories, measurement of the proper level of
stocks, and development of inventory and production control systems based
on the sorts of techniques described in this and following sections can be
very profitable. For example:
Johnson & Johnson has used these techniques for studying inventory require
ments for products with seasonally changing demand, and also to set economical
inventory goals balancing investment requirements against additional training
and overtime costs.
The American Thread Company, as a supplier to the fashion goods industry,
plagued with large inprocess inventories, daytoday imbalances among pro
duction departments, labor turnover, and customer service difficulties, found
these methods the key to improved scheduling and control procedures. Now
these improved procedures help keep an inventory of tens of thousands of items
in balance and smooth out production operations even in the face of demand
showing extremely erratic fluctuations due to fashion changes.
The Lamp Division of the General Electric Company has reported using
these methods to survey its finished inventory functions and stock requirements
in view of operating conditions and costs. This survey indicated how an im
proved warehouse reorder system would yield inventory cuts at both factories
and warehouses, and pointed to the reorder system characteristics that were
needed; it led to the installation of a new reorder and stock control system
offering substantial opportunities for stock reduction. The analytic approach
can also be used to show clearly what the cost in inventory investment and
schedule changes is to achieve a given level of customer service.
An industrial equipment manufacturer used these methods to investigate in
ventory and scheduling practices and to clear up policy ambiguities in this area,
as a prelude to installing an electronic computer system to handle inventory
control, scheduling, and purchase requisitions. In general, the analytic approach
has proved a valuable help in bringing disagreements over inventory policy into
the open, helping each side to recognize its own and the others' hidden assump
tions, and to reach a common agreement more quickly.
The Procter & Gamble Company recently described how analysis of its factory
inventory functions and requirements, using these methods, has pointed out
means for improved scheduling and more efficient use of finished stock. The
analysis indicated how the company could take advantage of certain particular
characteristics of its factories to cut stocks needed to meet sales fluctuations
GUIDES TO INVENTORY POLICY: I. FUNCTIONS AND LOT SIZES 181
while still maintaining its longstanding policy of guaranteed annual employ
ment.
These are only a few instances of applications. Numerous others could
be drawn from the experience of companies ranging from moderate to large
size, selling consumer goods or industrial products, with thousands of items
or only a few, and distribution in highly stable, predictable markets or in
erratically changing and unpredictable circumstances.
In the present article major attention will be devoted to (a) the con
ceptual framework of the analytic approach, including the definition of in
ventory function and the measurement of operational costs; and (b) the
problem of optimum lot size, with a detailed case illustration showing how
the techniques are applied.
This case reveals that the appropriate order quantity and the average in
ventory maintained do not vary directly with sales, and that a good answer
to the lot size question can be obtained with fairly crude cost data, pro
vided that a sound analytical approach is used. The case also shows that
the businessman does not need calculus to solve many inventory problems
(although use has to be made of it when certain complications arise).
INVENTORY PROBLEMS
The question before management is: How big should inventories be?
The answer to this is obvious — they should be just big enough. But what is
big enough?
This question is made more difficult by the fact that generally each indi
vidual within a management group tends to answer the question from his
own point of view. He fails to recognize costs outside his usual framework.
He tends to think of inventories in isolation from other operations. The
sales manager commonly says that the company must never make a cus
tomer wait; the production manager says there must be long manufacturing
runs for lower costs and steady employment; the treasurer says that large
inventories are draining off cash which could be used to make a profit.
Such a situation occurs all the time. The task of all production planning,
scheduling, or control functions, in fact, is typically to balance conflicting
objectives such as those of minimum purchase or production cost, minimum
inventory investment, minimum storage and distribution cost, and maximum
service to customers.
PRODUCTION VS. TIME
Often businessmen blame their inventory and scheduling difficulties on
small orders and product diversity: "You can't keep track of 100,000
182 INVENTORY CONTROL
items. Forecasts mean nothing. We're just a job shop." Many businessmen
seem to feel that their problems in this respect are unusual, whereas actually
the problems faced by a moderatesize manufacturer with a widely diversi
fied product line are almost typical of business today.
The fact is, simply, that under present methods of organization the costs
of paper work, setup, and control, in view of the diversity of products sold,
represent an extremely heavy drain on many a company's profit and a severe
cost to its customers. The superficial variety of output has often blinded
management to the opportunities for more systematic production flow and
for the elimination of many of the curses of jobshop operation by better
organization and planning.
The problem of planning and scheduling production or inventories per
vades all operations concerned with the matter of production versus time —
i.e., the interaction between production, distribution, and the location and
size of physical stocks. It occurs at almost every step in the production
process: purchasing, production of inprocess materials, finished production,
distribution of finished product, and service to customers. In multiplant
operations, the problem becomes compounded because decisions must be
made with reference to the amount of each item to be produced in each
factory; management must also specify how the warehouses should be
served by the plants.
ACTION VS. ANALYSIS
The questions businessmen raise in connection with management and
control of inventories are basically aimed at action, not at arriving at
answers. The questions are stated, unsurprisingly, in the characteristic
terms of decisions to be made: "Where shall we maintain how much
stock?" "Who will be responsible for it?" "What shall we do to control
balances or set proper schedules?" A manager necessarily thinks of prob
lems in production planning in terms of centers of responsibility.
However, action questions are not enough by themselves. In order to get
at the answers to these questions as a basis for taking action, it is necessary
to back off and ask some rather different kinds of questions: "Why do we
have inventories?" "What affects the inventory balances we maintain?"
"How do these effects take place?" From these questions, a picture of the
inventory problem can be built up which shows the influence on inventories
and costs of the various alternative decisions which the management may
ultimately want to consider.
This type of analytic or functional question has been answered intuitively
by businessmen with considerable success in the past. Consequently, most
of the effort toward improved inventory management has been spent in
other directions; it has been aimed at better means for recording, filing, or
GUIDES TO INVENTORY POLICY! I. FUNCTIONS AND LOT SIZES 183
displaying information and at better ways of doing the necessary clerical
work. This is all to the good, for efficient datahandling helps. However, it
does not lessen the need for a more systematic approach to inventory prob
lems that can take the place of, or at least supplement, intuition.
As business has grown, it has become more complex, and as business
executives have become more and more specialized in their jobs or farther
removed from direct operations, the task of achieving an economical bal
ance intuitively has become increasingly difficult. That is why more busi
nessmen are finding the concepts and mathematics of the growing field of
inventory theory to be of direct practical help.
One of the principal difficulties in the intuitive approach is that the types
and definitions of cost which influence appropriate inventory policy are not
those characteristically found on the books of a company. Many costs such
as setup or purchasing costs are hidden in the accounting records. Others
such as inventory capital costs may never appear at all. Each cost may be
clear to the operating head primarily responsible for its control; since it is
a "hidden" cost, however, its importance may not be clear at all to other
operating executives concerned. The resulting confusion may make it diffi
cult to arrive at anything like a consistent policy.
In the last five years in particular, operations research teams have suc
ceeded in using techniques of research scientists to develop a practical
analytic approach to inventory questions, despite growing business size,
complexity, and division of management responsibility.
INVENTORY FUNCTIONS
To understand the principles of the analytic approach, we must have
some idea of the basic functions of inventories.
Fundamentally, inventories serve to uncouple successive operations in
the process of making a product and getting it to consumers. For example,
inventories make it possible to process a product at a distance from cus
tomers or from raw material supplies, or to do two operations at a distance
from one another (perhaps only across the plant). Inventories make it un
necessary to gear production directly to consumption or, alternatively, to
force consumption to adapt to the necessities of production. In these and
similar ways, inventories free one stage in the productiondistribution
process from the next, permitting each to operate more economically.
The essential question is : At what point does the uncoupling function of
inventory stop earning enough advantage to justify the investment required?
To arrive at a satisfactory answer we must first distinguish between (a) in
ventories necessary because it takes time to complete an operation and to
move the product from one stage to another; and (b) inventories employed
184 INVENTORY CONTROL
for organizational reasons, i.e., to let one unit schedule its operations more
or less independently of another.
MOVEMENT INVENTORIES
Inventory balances needed because of the time required to move stocks
from one place to another are often not recognized, or are confused with
inventories resulting from other needs — e.g., economical shipping quantities
(to be discussed in a later section).
The average amount of movement inventory can be determined from the
mathematical expression / = S X T in which S represents the average sales
rate, T the transit time from one stage to the next, and / the movement in
ventory needed. For example, if it takes two weeks to move materials from
the plant to a warehouse, and the warehouse sells 100 units per week, the
average inventory in movement is 100 units per week times 2 weeks, or
200 units. From a different point of view, when a unit is manufactured and
ready for use at the plant, it must sit idle for two weeks while being moved
to the next station (the warehouse); so, on the average, stocks equal to
two weeks' sales will be in movement.
Movement inventories are usually thought of in connection with move
ment between distant points — plant to warehouse. However, any plant may
contain substantial stocks in movement from one operation to another —
for example, the product moving along an assembly line. Movement stock
is one component of the "float" or inprocess inventory in a manufacturing
operation.
The amount of movement stock changes only when sales or the time in
transit is changed. Time in transit is largely a result of method of trans
portation, although improvements in loading or dispatching practices may
cut transit time by eliminating unnecessary delays. Other somewhat more
subtle influences of time in transit on total inventories will be described in
connection with safety stocks.
ORGANIZATION INVENTORIES
Management's most difficult problems are with the inventories that "buy"
organization in the sense that the more of them management carries be
tween stages in the manufacturingdistribution process, the less coordination
is required to keep the process running smoothly. Contrariwise, if inven
tories are already being used efficiently, they can be cut only at the expense
of greater organization effort — e.g., greater scheduling effort to keep succes
sive stages in balance, and greater expediting effort to work out of the diffi
culties which unforeseen disruptions at one point or another may cause in
the whole process.
GUIDES TO INVENTORY POLICY: I. FUNCTIONS AND LOT SIZES 185
Despite superficial differences among businesses in the nature and char
acteristics of the organization inventory they maintain, the following three
functions are basic:
(1) Lot size inventories are probably the most common in business. They
are maintained wherever the user makes or purchases material in larger lots
than are needed for his immediate purposes. For example, it is common prac
tice to buy raw materials in relatively large quantities in order to obtain quan
tity price discounts, keep shipping costs in balance, and hold down clerical costs
connected with making out requisitions, checking receipts, and handling ac
counts payable. Similar reasons lead to long production runs on equipment
calling for expensive setup, or to sizable replenishment orders placed on fac
tories by field warehouses.
(2) Fluctuation stocks, also very common in business, are held to cushion
the shocks arising basically from unpredictable fluctuations in consumer de
mand. For example, warehouses and retail outlets maintain stocks to be able to
supply consumers on demand, even when the rate of consumer demand may
show quite irregular and unpredictable fluctuations. In turn, factories maintain
stocks to be in a position to replenish retail and field warehouse stocks in line
with customer demands.
Shortterm fluctuations in the mix of orders on a plant often make it neces
sary to carry stocks of parts of subassemblies, in order to give assembly opera
tions' flexibility in meeting orders as they arise while freeing earlier operations
(e.g., machining) from the need to make momentary adjustments in schedules
to meet assembly requirements. Fluctuation stocks may also be carried in semi
finished form in order to balance out the load among manufacturing depart
ments when orders received during the current day, week, or month may put a
load on individual departments which is out of balance with longrun require
ments.
In most cases, anticipating all fluctuations is uneconomical, if not impossible.
But a business cannot get along without some fluctuation stocks unless it is
willing and able always to make its customers wait until the material needed can
be purchased conveniently or until their orders can be scheduled into produc
tion conveniently. Fluctuation stocks are part of the price we pay for our gen
eral business philosophy of serving the consumers' wants (and whims!) rather
than having them take what they can get. The queues before Russian retail
stores illustrate a different point of view.
(3) Anticipation stocks are needed where goods or materials are consumed
on a predictable but changing pattern through the year, and where it is desir
able to absorb some of these changes by building and depleting inventories
rather than by changing production rates with attendant fluctuations in employ
ment and additional capital capacity requirements. For example, inventories
may be built up in anticipation of a special sale or to fill needs during a plant
shutdown.
The need for seasonal stocks may also arise where materials (e.g., agricultural
products) are produced at seasonally fluctuating rates but where consumption
186 INVENTORY CONTROL
is reasonably uniform; here the problems connected with producing and storing
tomato catsup are a prime example. 1
STRIKING A BALANCE
The joker is that the gains which these organization inventories achieve
in the way of less need for coordination and planning, less clerical effort to
handle orders, and greater economies in manufacturing and shipping are not
in direct proportion to the size of inventory. Even if the additional stocks
are kept well balanced and properly located, the gains become smaller,
while at the same time the warehouse, obsolescence, and capital costs asso
ciated with maintaining inventories rise in proportion to, or perhaps even
at a faster rate than, the inventories themselves. To illustrate:
Suppose a plant needs 2,000 units of a specially machined part in a year. If
these are made in runs of 100 units each, then 20 runs with attendant setup
costs will be required each year.
If the production quantity were increased from 100 to 200 units, only 10
runs would be required — a 50% reduction in setup costs, but a 100% increase
in the size of a run and in the resulting inventory balance carried.
If the runs were further increased in length to 400 units each, only 5 produc
tion runs during the year would be required — only 25% more reduction in setup
costs, but 200% more increase in run length and inventory balances.
The basic problem of inventory policy connected with the three types of
inventories which "buy" organization is to strike a balance between the
increasing costs and the declining return earned from additional stocks.
It is because striking this balance is easier to say than to do, and because
it is a problem that defies solution through an intuitive understanding
alone, that the new analytical concepts are necessary.
INVENTORY COSTS
This brings us face to face with the question of the costs that influence
inventory policy, and the fact, noted earlier, that they are characteristically
not those recorded, at least not in directly available form, in the usual in
dustrial accounting system. Accounting costs are derived under principles
developed over many years and strongly influenced by tradition. The specific
methods and degree of skill and refinement may be better in particular
companies, but in all of them the basic objective of accounting procedures
is to provide a fair, consistent, and conservative valuation of assets and a
picture of the flow of values in the business.
In contrast to the principles and search for consistency underlying ac
1 See Alexander Henderson and Robert Schlaifer, "Mathematical Programming:
Better Information for Better DecisionMaking," reprinted here on p. 53.
GUIDES TO INVENTORY POLICY: I. FUNCTIONS AND LOT SIZES 187
counting costs, the definition of costs for production and inventory control
will vary from time to time — even in the same company — according to the
circumstances and the length of the period being planned for. The following
criteria apply:
( 1 ) The costs shall represent "outofpocket" expenditures, i.e., cash actually
paid out or opportunities for profit foregone. Overtime premium payments are
outofpocket; depreciation on equipment on hand is not. To the extent that
storage space is available and cannot be used for other productive purposes, no
outofpocket cost of space is incurred; but to the extent that storage space is
rented (outofpocket) or could be used for other productive purposes (fore
gone opportunity), a suitable charge is justified. The charge for investment is
based on the outofpocket investment in inventories or added facilities, not on
the "book" or accounting value of the investment.
The rate of interest charged on outofpocket investment may be based either
on the rate paid banks (outofpocket) or on the rate of profit that might
reasonably be earned by alternative uses of investment (foregone opportunity),
depending on the financial policies of the business. In some cases, a bank rate
may be used on shortterm seasonal inventories and an internal rate for long
term, minimum requirements.
Obviously, much depends on the time scale in classifying a given item. In
the short run, few costs are controllable outofpocket costs; in the long run,
all are.
(2) The costs shall represent only those outofpocket expenditures or fore
gone opportunities for profit whose magnitude is affected by the schedule or
plan. Many overhead costs, such as supervision costs, are outofpocket, but
neither the timing nor the size is affected by the schedule. Normal material and
direct labor costs are unaffected in total and so are not considered directly;
however, these as well as some components of overhead cost do represent out
ofpocket investments, and accordingly enter the picture indirectly through any
charge for capital.
DIRECT INFLUENCE
Among the costs which directly influence inventory policy are (a) costs
depending on the amount ordered, (b) production costs, and (c) costs of
storing and handling inventory.
Costs that depend on the amount ordered — These include, for example,
quantity discounts offered by vendors; setup costs in internal manufacturing
operations and clerical costs of making out a purchase order; and, when capacity
is pressed, the profit on production lost during downtime for setup. Shipping
costs represent another factor to the extent that they influence the quantity of
raw materials purchased and resulting raw stock levels, the size of intraplant or
plantwarehouse shipments, or the size and the frequency of shipments to
customers.
Production costs — Beyond setup or changeover costs, which are included in
188 INVENTORY CONTROL
the preceding category, there are the abnormal or nonroutine costs of produc
tion whose size may be affected by the policies or control methods used.
(Normal or standard raw material and direct labor costs are not significant in
inventory control; these relate to the total quantity sold rather than to the
amount stocked.) Overtime, shakedown, hiring, and training represent costs
that have a direct bearing on inventory policy.
To illustrate, shakedown or learning costs show up wherever output during
the early part of a new run is below standard in quantity or quality. 2 A cost of
undercapacity operation may also be encountered — for example, where a basic
labor force must be maintained regardless of volume (although sometimes this
can be looked on as part of the fixed facility cost, despite the fact that it is
accounted for as a directly variable labor cost).
Costs of handling and storing inventory — In this group of costs affected by
control methods and inventory policies are expenses of handling products in
and out of stock, storage costs such as rent and heat, insurance and taxes,
obsolescence and spoilage costs, and capital costs (which will receive detailed
examination in the next section).
Inventory obsolescence and spoilage costs may take several forms, including
(1) outright spoilage after a more or less fixed period; (2) risk that a particular
unit in stock or a particular product number will (a) become technologically
unsalable, except perhaps at a discount or as spare parts, (b) go out of style,
or (c) spoil.
Certain food and drug products, for example, have specified maximum shelf
lives and must either be used within a fixed period of time or be dumped.
Some kinds of style goods, such as many lines of toys, Christmas novelties, or
women's clothes, may effectively "spoil" at the end of a season, with only re
claim or dump value. Some kinds of technical equipment undergo almost
constant engineering change during their production life; thus component stocks
may suddenly and unexpectedly be made obsolete.
CAPITAL INVESTMENT
Evaluating the effect of inventory and scheduling policy upon capital
investment and the worth of capital tied up in inventories is one of the most
difficult problems in resolving inventory policy questions.
Think for a moment of the amount of capital invested in inventory. This
is the outofpocket, or avoidable, cash cost for material, labor, and over
head of goods in inventory (as distinguished from the "book" or account
ing value of inventory). For example, raw materials are normally purchased
in accordance with production schedules; and if the production of an item
can be postponed, buying and paying for raw materials can likewise be
put off.
2 See Frank J. Andress, "The Learning Curve as a Production Tool," HBR Janu
aryFebruary 1954, p. 87.
GUIDES TO INVENTORY POLICY: I. FUNCTIONS AND LOT SIZES 189
Usually, then, the raw material cost component represents a part of the
outofpocket inventory investment in finished goods. However, if raw
materials must be purchased when available (e.g., agricultural crops)
regardless of the production schedule, the raw material component of
finished product cost does not represent avoidable investment and therefore
should be struck from the computation of inventory value for planning
purposes.
As for maintenance and similar factory overhead items, they are usually
paid for the year round, regardless of the timing of production scheduled;
therefore these elements of burden should not be counted as part of the
product investment for planning purposes. (One exception: if, as sometimes
happens, the maintenance costs actually vary directly with the production
rate as, for example, in the case of supplies, they should of course be
included. )
Again, supervision, at least general supervision, is usually a fixed monthly
cost which the schedule will not influence, and hence should not be in
cluded. Depreciation is another type of burden item representing a charge
for equipment and facilities already bought and paid for; the timing of the
production schedule cannot influence these past investments and, while they
represent a legitimate cost for accounting purposes, they should not be
counted as part of the inventory investment for inventory and production
planning purposes.
In sum, the rule is this : for production planning and inventory manage
ment purposes, the investment value of goods in inventory should be taken
as the cash outlay made at the time of production that could have been
delayed if the goods were not made then but at a later time, closer to the
time of sale.
Cost of Capital Invested. This item is the product of three factors:
(a) the capital value of a unit of inventory, (b) the time a unit of product
is in inventory, and (c) the charge or imputed interest rate placed against
a dollar of invested cash. The first factor was mentioned above. As for the
second, it is fixed by management's inventory policy decisions. But these
decisions can be made economically only in view of the third factor. This
factor depends directly on the financial policy of the business.
Sometimes businessmen make the mistake of thinking that cash tied up
in inventories costs nothing, especially if the cash to finance inventory is
generated internally through profits and depreciation. However, this implies
that the cash in inventories would otherwise sit idle. In fact, the cash could,
at least, be invested in government bonds if not in inventories. And if it
were really idle, the cash very likely should be released to stockholders for
profitable investment elsewhere.
190
INVENTORY CONTROL
Moreover, it is dangerous to assume that, as a "shortterm" investment,
inventory is relatively liquid and riskless. Businessmen say, "After all, we
turn our inventory investment over six times a year." But, in reality, inven
tory investment may or may not be shortterm and riskless, depending on
circumstances. No broad generalization is possible, and each case must be
decided on its own merits. For example:
A great deal of inventory carried in business is as much a part of the perma
nent investment as the machinery and buildings. The inventory must be main
tained to make operations possible as long as the business is a going concern.
The cash investment released by the sale of one item from stock must be
promptly reinvested in new stock, and the inventory can be liquidated only
when the company is closed. How much more riskless is this than other fixed
manufacturing assets?
To take an extreme case, inventory in fashion lines or other types of products
having high obsolescence carries a definite risk. Its value depends wholly on the
company's ability to sell it. If sales are insufficient to liquidate the inventory
built up, considerable losses may result.
At the other extreme, inventory in stable product lines built up to absorb
shortterm seasonal fluctuations might be thought of as bearing the least risk,
since this type of investment is characteristically shortterm. But even in these
cases there can be losses. Suppose, for instance, that peak seasonal sales do not
reach anticipated levels and substantially increased costs of storage and obso
lescence have to be incurred before the excess inventory can be liquidated.
Finally, it might be pointed out that the cost of the dollars invested in
inventory may be underestimated if bank interest rate is used as the basis,
ignoring the riskbearing or entrepreneur's compensation. How many busi
nessmen are actually satisfied with uses of their companies' capital funds
which do not earn more than a lender's rate of return? In choosing a truly
appropriate rate — a matter of financial policy — the executive must answer
some questions :
1. Where is the cash coming from — inside earnings or outside financing?
2. What else could we do with the funds, and what could we earn?
3. When can we get the investment back out, if ever?
4. How much risk of sales disappointment and obsolescence is really connected
with this inventory?
5. How much of a return do we want, in view of what we could earn elsewhere
or in view of the cost of money to us and the risk the inventory investment
entails?
Investment in Facilities. Valuation of investment in facilities is generally
important only in longrun planning problems — as, for example, when in
creases in productive or warehouse capacity are being considered. (Where
facilities already exist and are not usable for other purposes, and where
GUIDES TO INVENTORY POLICY: I. FUNCTIONS AND LOT SIZES 191
planning or scheduling do not contemplate changing these existing facilities,
investment is not affected.)
Facilities investment may also be important where productive capacity is
taxed, and where the form of the plan or schedule will determine the amount
of added capacity which must be installed, either to meet the plan itself or
for alternative uses. In such cases, considerable care is necessary in defining
the facilities investment in order to be consistent with the principles noted
above: i.e., that facilities investment should represent outofpocket invest
ment, or, alternatively, foregone opportunities to make outofpocket in
vestment elsewhere.
CUSTOMER SERVICE
An important objective in most production planning and inventory con
trol systems is maintenance of reasonable customer service. An evaluation
of the worth of customer service, or the loss suffered through poor service,
is an important part of the problem of arriving at a reasonable inventory
policy. This cost is typically very difficult to arrive at, including as it does
the paper work costs of rehandling back orders and, usually much more
important, the effect that dissatisfaction of customers may have on future
profits.
In some cases it may be possible to limit consideration to the cost of
producing the needed material on overtime or of purchasing it from the
outside and losing the contribution to profit which it would have made. On
the other hand, sometimes the possible loss of customers and their sales
over a substantial time may outweigh the cost of direct loss in immediate
business, and it may be necessary to arrive at a statement of a "reasonable"
level of customer service — i.e., the degree of risk of running out of stock,
or perhaps the number of times a year the management is willing to run
out of an item. In other cases, it may be possible to arrive at a reasonable
maximum level of sales which the company is prepared to meet with 100%
reliability, being reconciled to have service suffer if sales exceed this level.
One of the uses of the analytic techniques described below and in follow
ing parts of this series is to help management arrive at a realistic view of
the cost of poor service, or of the value of building high service, by laying
out clearly what the cost in inventory investment and schedule changes is
to achieve this degree of customer service. Sometimes when these costs are
clearly brought home, even a 100% serviceminded management is willing
to settle for a more realistic, "excellent" service at moderate cost, instead
of striving for "perfect" service entailing extreme cost.
192 INVENTORY CONTROL
OPTIMUM LOT SIZE
Now, with this background, let us examine in some detail one of the
inventory problems which plague businessmen the most — that of the opti
mum size of lot to purchase or produce for stock. This happens also to be
one of the oldest problems discussed in the industrial engineering texts —
but this does not lessen the fact that it is one of the most profitable for a
great many companies to attack today with new analytic techniques.
COMMON PRACTICES
This problem arises, as mentioned earlier, because of the need to purchase
or produce in quantities greater than will be used or sold. Thus, specifically,
businessmen buy raw materials in sizable quantities — carloads, or even
trainloads — in order to reduce the costs connected with purchasing and
control, to obtain a favorable price, and to minimize handling and trans
portation costs. They replenish factory inprocess stocks of parts in sizable
quantities to avoid, where possible, the costs of equipment setups and
clerical routines. Likewise, finished stocks maintained in warehouses usu
ally come in shipments substantially greater than the typical amount sold at
once, the motive again being, in part, to avoid equipment setup and paper
work costs and, in the case of field warehouses, to minimize shipping costs.
Where the same equipment is used for a variety of items, the equipment
will be devoted first to one item and then to another in sequence, with the
length of the run in any individual item to be chosen, as far as is economi
cally possible, to minimize changeover cost from one item to another and
to reduce the production time lost because of cleanout requirements during
changeovers. Blocked operations of this sort are seen frequently, for ex
ample, in the petroleum industry, on packaging lines, or on assembly lines
where changeover from one model to another may require adjustment in
feed speeds and settings and change of components.
In all these cases, the practice of replenishing stocks in sizable quantities
compared with the typical usage quantity means that inventory has to be
carried; it makes it possible to spread fixed costs (e.g., setup and clerical
costs) over many units and thus to reduce the unit cost. However, one can
carry this principle only so far, for if the replenishment orders become too
large, the resulting inventories get out of line, and the capital and handling
costs of carrying these inventories more than offset the possible savings in
production, transportation, and clerical costs. Here is the matter, again, of
striking a balance between these conflicting considerations.
Even though formulas for selecting the optimum lot size are presented in
GUIDES TO INVENTORY POLICY! I. FUNCTIONS AND LOT SIZES 193
many industrial engineering texts, 3 few companies make any attempt to
arrive at an explicit quantitative balance of inventory and changeover or
setup costs. Why?
For one thing, the cost elements which enter into an explicit solution fre
quently are very difficult to measure, or are only very hazily defined. For
example, it may be possible to get a fairly accurate measure of the cost of
setting up a particular machine, but it may be almost impossible to derive
a precise measure of the cost of making out a new production order. Again,
warehouse costs may be accumulated separately on the accounting records,
but these rarely show what the cost of housing an additional unit of material
may be. In my experience the capital cost, or imputed interest cost, con
nected with inventory investment never appears on the company's account
ing records.
Furthermore, the inventory is traditionally valued in such a way that the
true incremental investment is difficult to measure for scheduling purposes.
Often, companies therefore attempt to strike only a qualitative balance
of these costs to arrive at something like an optimum or minimumcost
reorder quantity.
Despite the difficulty in measuring costs — and indeed because of such
difficulty — it is eminently worthwhile to look at the lot size problem ex
plicitly formulated. The value of an analytic solution does not rest solely on
one's ability to plug in precise cost data to get an answer. An analytic solu
tion often helps clarify questions of principle, even with only crude data
available for use. Moreover, it appears that many companies today still
have not accepted the philosophy of optimum reorder quantities from the
overall company standpoint; instead, decisions are dominated from the
standpoint of some particular interest such as production or traffic and
transportation. Here too the analytic solution can be of help, even when the
cost data are incomplete or imperfect.
CASE EXAMPLE
To illustrate how the lot size problem can be attacked analytically — and
what some of the problems and advantages of such an attack are — let us
take a fictitious example. The situation is greatly oversimplified on purpose
to get quickly to the heart of the analytic approach.
Elements of the Problem. Brown and Brown, Inc., an automotive parts
3 See, for example, Raymond E. Fairfield, Quantity and Economy in Manufacture
(New York, D. Van Nostrand Company, Inc., 1931).
194
INVENTORY CONTROL
supplier, produces a simple patented electric switch on longterm contracts.
The covering is purchased on the outside at $0.01 each, and 1,000 are used
regularly each day, 250 days per year.
The casings are made in a nearby plant, and B. and B. sends its own
truck to pick them up. The cost of truck operation, maintenance, and the
driver amounts to $10 per trip.
The company can send the truck once a day to bring back 1 ,000 casings
for that day's requirements, but this makes the cost of a casing rather high.
The truck can go less frequently, but this means that it has to bring back
more than the company needs for its immediate daytoday purposes.
The characteristic "sawtooth" inventory pattern which will result is
shown in Exhibit I, where 1,000 Q casings are picked up each trip (Q being
whatever number of days' supply is obtained per replenishment trip) . These
are used up over a period of Q days. When the inventory is depleted again,
another trip is made to pick up Q days' supply or 1,000 Q casings once
more, and so on.
EXHIBIT I. PATTERN OF INVENTORY BALANCE
(1,000 Q casings obtained per replenishment trip; 1,000 casings used per day)
1,000 o
Inventory
of casings
•* — days ^\
Time
B. and B. estimates that the cost of storing casings under properly con
trolled humidity conditions is $1 per 1,000 casings per year. The company
wants to obtain a 10% return on its inventory investment of $10 (1,000
times $0.01), which means that it should properly charge an additional $1
(10% of $10), making a total inventory cost of $2 per 1,000 casings per
year.
(Note that, in order to avoid undue complications, the inventory invest
ment charge is made here only against the purchase price of the casings and
not against the total delivery cost including transportation. Where trans
portation is a major component of total cost, it is of course possible and
desirable to include it in the base for the inventory charge.)
GUIDES TO INVENTORY POLICY: I.. FUNCTIONS AND LOT SIZES
195
Graphic Solution. Brown and Brown, Inc., can find what it should do by
means of a graph (see Exhibit II) showing the annual cost of buying,
moving, and storing casings:
The broken line shows total trucking costs versus the size of the indi
vidual purchase quantity:
EXHIBIT II. ANNUAL COST OF BUYING, MOVING, AND STORING
CASINGS COMPARED WITH REORDER QUANTITY
300
250
200
150
too
50
Inventory and
storage cost
Minimum cost quantity
_j
20,000 40,000 60,000 80,000 100,000
Quantity of casings obtained each trip
If 1,000 casings are purchased at a time, the total cost is $10 times 25(
trips, or $2,500 per year.
If 10,000 casings are purchased at one time, only 25 trips need be made,
for a total cost of $250 per year.
If 100,000 casings are purchased, only IVi trips, on the average, have to
be taken each year, for a total cost of $25.
The dotted line shows the inventory cost compared with the size of the pur
chased quantity:
If 10,000 casings are purchased at one time, the inventory at purchase will
contain 10,000, and it will gradually be depleted until none are on hand,
when a new purchase will be made. The average inventory on hand thus
will be 5,000 casings. The cost per year will be $2 times 5,000 casings, or
$10.
196 INVENTORY CONTROL
EXHIBIT III. EXAMPLE OF ALGEBRAIC SOLUTION OF SAME
INVENTORY PROBLEM AS EXHIBIT II
The total annual cost of supplying casings is equal to the sum of the direct cost of
the casings, plus the trucking cost, plus the inventory and storage cost.
Let:
T = total annual cost
b — unit purchase price, $10 per 1,000 casings
s = annual usage, 250,000 casings
A = trucking cost, $10 per trip
N = number of trips per year
i — cost of carrying casings in inventory at the annual rate of $2 per 1,000, or
$0,002 per casing
x = size of an individual purchase (x/2 = average inventory)
Then the basic equation will be:
T = bs + AN + ix/2
The problem is to choose the minimumcost value of x (or, if desired, N). Since x
is the same as s/N, N can be expressed as s/x. Substituting s/x for N in the above
equation, we get:
T — bs + As/x + ix/2
From this point on we shall use differential calculus. The derivative of total cost, T,
with respect to x will be expressed as:
dT/dx — — As/x 2 + i/2
And the minimumcost value of x is that for which the derivative of total cost with
respect to x equals zero. This is true when:
x = y/2As/i
Substituting the known values for A, s, and i:
x = V210250,000/.002 = 50,000 casings
Similarly, if 100,000 casings are purchased at one time, the average in
ventory will be 50,000 casings, and the total inventory and storage cost
will be $100.
The solid line is the total cost, including both trucking and inventory and
storage costs. The total cost is at a minimum when 50,000 casings are
purchased on each trip and 5 trips are made each year, for at this point the
total trucking cost and the total inventory and storage cost are equal.
The solution to B. and B.'s problem can be reached algebraically as well
as graphically. Exhibit III shows how the approach works in this very
simple case.
SIMILAR CASES
The problem of Brown and Brown, Inc., though artificial, is not too far
from the questions many businesses face in fixing reorder quantities.
GUIDES TO INVENTORY POLICY: I. FUNCTIONS AND LOT SIZES
197
Despite the simplifications introduced — for example, the assumption that
usage is known in advance — the method of solution has been found widely
useful in industries ranging from mail order merchandising (replenishing
staple lines), through electrical equipment manufacturing (ordering ma
chined parts to replenish stockrooms), to shoe manufacturing (ordering
findings and other purchased supplies). In particular, the approach has
been found helpful in controlling stocks made up of many lowvalue items
used regularly in large quantities.
EXHIBIT IV.
Q[\S/P)
INFLUENCE OF PRODUCTION AND SALES RATE ON
PRODUCTION CYCLE INVENTORY
Inventory of casings
Time
A number of realistic complications might have been introduced into the
Brown and Brown, Inc., problem. For example:
In determining the size of a manufacturing run, it sometimes is important
to acccount explicitly for the production and sales rate. In this case, the in
ventory balance pattern looks like Exhibit IV instead of the sawtooth design
in Exhibit I, The maximum inventory point is not equal to the amount
produced in an individual run, but to that quantity less the amount sold during
the course of the run. The maximum inventory equals Q (1 — S/P), where
Q is the amount produced in a single run, and S and P are the daily sales and
production rates respectively.
This refinement can be important, particularly if the sales rate is fairly
large compared with the production rate. Thus, if the sales rate is half the
production rate, then the maximum inventory is only half the quantity made
in one run, and the average inventory equals only onefourth the individual
run quantity. This means that substantially more inventory can be carried —
in fact, about 40% more.
When a number of products are made on a regular cycle, one after another,
with the sequence in the cycle established by economy in changeover cost,
the total cycle length can be obtained in the same way as described above.
Of course, it sometimes happens that there is a periodic breach in the cycle,
either to make an occasional run of a product with very low sales or to allow
for planned maintenance of equipment; the very simple runlength formulas
can be adjusted to allow for this.
Other kinds of costs can also be included, such as different sorts of handling
costs. Or the inventory cost can be defined in such a way as to include trans
198
INVENTORY CONTROL
portation, obsolescence, or even capital and storage cost as part of the unit
value of the product against which a charge for capital is made. When a
charge for capital is included as part of the base value in computing the cost
of capital, this is equivalent to requiring that capital earnings be compounded;
this can have an important bearing on decisions connected with very low volume
items which might be purchased in relatively large, longlasting quantities.
Complications such as the foregoing, while important in practice, repre
sent changes in arithmetic rather than in basic concept.
SIGNIFICANT CONCLUSIONS
When the analytic approach is applied to Brown and Brown's problem
and similar cases, it reveals certain relationships which are significant and
useful to executives concerned with inventory management:
(1) The appropriate order quantity and the average inventory maintained
do not vary directly with sales. In fact, both of these quantities vary with the
square root of sales. This means that with the same ordering and setup cost
characteristics, the larger the volume of sales of an item, the less inventory per
unit of sales is required. One of the sources of inefficiency in many inventory
control systems is the rigid adoption of a rule for ordering or carrying in
ventory equivalent to, say, one month's sales.
(2) The total cost in the neighborhood of the optimum order quantity is
relatively insensitive to moderately small changes in the amount ordered. Ex
hibit II illustrates this proposition. Thus, all that is needed is just to get in the
"right ball park," and a good answer can be obtained even with fairly crude
cost data. For example, suppose the company had estimated that its total
cost of holding 1,000 casings in inventory for a year was $1 when it actually
was $2 (as in our illustration). Working through the same arithmetic, the
company would have arrived at an optimum order quantity of 70,000 casings
instead of 50,000. Even so, the total cost would have been (using the correct
$2 annual carrying cost) :
3.6 trips per year @ $10 = $36
35,000 casings average inventory @ $0,002 = 70
Total annual cost = $106
Thus, an error of a factor of 2 in one cost results in only a 6% difference
in total cost.
In summary, Brown and Brown's problem, despite its oversimplification,
provides an introduction to the analytic approach to inventory problems.
In particular, it illustrates the first essential in such an approach — i.e.,
defining an inventory function. In this case the function is to permit pur
chase or manufacture in economical order quantities or run lengths; in other
cases it may be different. The important point is that this basic function can
GUIDES TO INVENTORY POLICY: I. FUNCTIONS AND LOT SIZES 199
be identified wherever it may be found — in manufacturing, purchasing, or
warehouse operation.
The only way to cut inventories is to organize operations so that they are
tied more closely together. For example, a company can cut its raw
materials inventory by buying in smaller quantities closer to needs, but it
does so at a cost; this cost results from the increased clerical operations
needed to tie the purchasing function more closely to manufacturing and
to keep it more fully informed of manufacturing's plans and operations. The
right inventory level is reached when the cost of maintaining any additional
inventory custion offsets the saving that the additional inventory earns by
permitting the plant to operate in a somewhat less fully organized fashion.
B. and B.'s problem also illustrates problems and questions connected
with defining and making costs explicit. The inventory capital cost is usually
not found on a company's books, but it is implied in some of the disagree
ments over inventory policy. Here, again, bringing the matter into the open
may help each side in a discussion to recognize its own and the others'
hidden assumptions, and thus more quickly to reach a common agreement.
Chapter 10
UNCERTAINTY PROBLEMS
IN INVENTORY CONTROL
XII.
Guides to Inventory Policy
II. Problems of Uncertainty*
John F. Magee
Marketing and production executives alike have an immediate, vital interest
in safety stocks. In these days of strong but often unpredictable sales,
safety stocks afford, for the factory as well as for the sales office, a method
of buying shortterm protection against the uncertainties of customer
demand. They are the additional inventory on hand which can be drawn
upon in case of emergency during the period between placement of an
order by the customer and receipt of the material to fill the order. However,
in practice their potentials are often needlessly lost.
One reason for the failure is a very practical one. Because safety stocks
are designed to cope with the uncertainties of sales, they must be controlled
by flexible rules so that conditions can be met as they develop. But some
times the need for flexibility is used as an excuse for indefiniteness : "We
can't count on a thing; we have to play the situation by ear." And, in any
* From the Harvard Business Review, Vol. 34, No. 2 (1956), 103116. Reprinted
by permission of the Harvard Business Review.
200
GUIDES TO INVENTORY POLICY! II. PROBLEMS OF UNCERTAINTY 201
sizable organization, when people at the factory level start "playing it by
ear," one can be almost sure that management policy will not be regularly
translated into practice.
Our studies have shown that the methods used by existing systems in
industry often violate sound control concepts. The economy of the com
pany is maintained, in the face of instability and inefficiency in the in
ventory control system, only because of constant attention, exercise of
overriding common sense, and use of expediting and other emergency meas
ures outside the routine of the system.
Actually, it is possible to have inventory controls which are not only
flexible but also carefully designed and explicit. But the task needs special
analytical tools; in a complicated business it defies commonsense judg
ment and simple arithmetic. Methods must be employed to take direct
account of uncertainty and to measure the response characteristics of the
system and relate them to costs. Such methods are the distinctive mark of
a really modern, progressive inventory control system.
Here are some of the points which I shall discuss in this article:
Basically, there are two different types of inventory replenishment systems
designed to handle uncertainty about sales — fixed order, commonly used in
stockrooms and factories, as in bins of parts or other materials; and periodic
reordering, frequently used in warehouses for inventories involving a large
number of items under clerical control. While the two are basically similar in
concept, they have somewhat different effects on safety stocks, and choice of
one or the other, or some related variety, requires careful consideration. Cer
tain factors which should be taken into account in the choice between them
will be outlined.
The fundamental problem of setting safety stocks under either system is
balancing a series of types of costs which are not found in the ordinary ac
counting records of the company — costs of customer service failure, of varying
production rates (including hiring and training expenses), of space capacity,
and others. Often specialists can find the optimum balance with relatively simple
techniques once the cost data are made explicit. However, part of the needed
data can come only from top management. For example, the tolerable risk
of service failure is generally a policy decision.
The specific problem of inventory control, including production scheduling,
varies widely from company to company. Where finished items can be stocked,
the important cost factors to weigh may be storage, clerical procedures, setup,
supervision, etc. But where finished items cannot be stocked, the problem is
one of setting capacity levels large enough to handle fluctuating loads without
undue delay, which involves the cost of unused labor and machines. Despite
the great variety of situations that are possible, specific mathematical approaches
and theories are available for use in solving almost any type of company prob
lem.
Both to illustrate the various techniques and by way of summary, a hypo
202 INVENTORY CONTROL
thetical case will be set forth where a company moved through a series of
stages of inventory control. Significantly, the final step brought a large re
duction in stocks needed for efficient service and also a great reduction in
production fluctuations. Out of the range of this company's experience, other
managements should be able to get some guidance as to what is appropriate
for their own situations.
BASIC SYSTEMS
Like transit stocks and lotsize stocks (discussed specifically in the
preceding article in this series) and also anticipation stocks (to be taken
up in the subsequent article), safety stocks "decouple" one stage in pro
duction and distribution from the next, reducing the amount of overall
organization and control needed.
But the economies of safety inventories are not fairly certain and
immediate. The objective is to arrive at a reasonable balance between the
costs of the stock and the protection obtained against inventory exhaustion.
Since exhaustion becomes less likely as the safety inventory increases, each
additional amount of safety inventory characteristically buys relatively
less protection. The return from increasing inventory balances therefore
diminishes rapidly. So the question is: How much additional inventory as
safety stock can be economically justified?
To answer this question we need to look at the two basic systems of
inventory replenishment to handle uncertainty about sales and see how
they produce different results.
FIXED ORDER
Under any fixed order system — the oldfashioned "twobin" system
or one of its modern varieties — the same quantity of material is always
ordered (a binful in the primitive system), but the time an order is placed
is allowed to vary with fluctuations in usage (when the bottom of one bin
is reached). The objective is to place an order whenever the amount on
hand is just sufficient to meet a "reasonable" maximum demand over the
course of the lead time which must be allowed between placement of the
replenishment order and receipt of the material.
Where the replenishment lead time is long (e.g., three months) com
pared with the amount purchased at each order (e.g., a onemonth sup
ply), there are presumably some purchase orders outstanding all the time
which, on being filled, will help replenish the existing inventory on hand.
In such cases, of course, the safety stocks and reorder points should be
based upon both amount on hand and on order. Where, on the other hand,
the lead time is short compared with the quantity ordered, as in most
GUIDES TO INVENTORY POLICY: II. PROBLEMS OF UNCERTAINTY
203
factory twobin systems, the amount on hand and the total on hand and
on order are in fact equivalent at the time of reordering.
The key to setting the safety stock is the "reasonable" maximum usage
during the lead time. What is "reasonable" depends partly, of course, on
the nature of shortterm fluctuations in the rate of sale. It also depends —
and here is where the top executive comes foremost into the picture —
on the risk that management is prepared to face in running out of stock.
What is the level of sales or usage beyond which management is prepared
to face the shortages? For example :
In Exhibit I, continuing the hypothetical case of Brown and Brown, Inc.,
discussed in the first article of this series, the curve shows the number of
weeks in which the demand for casings may be expected to equal or exceed
any specified level. (Such a curve could be roughly plotted according to
actual experience modified by such expectations or projections as seem war
ranted; refinement can be added by the use of mathematical analysis when
such precision seems desirable.)
EXHIBIT I. BROWN AND BROWN'S SAFETY STOCK
Percentage of time demand exceeds Level D
100°/ e
10 20 30 40 50 60 70 80
Level of demand (thousands of units)
204 INVENTORY CONTROL
Now, if it takes B. and B. a week to replenish its stocks and the manage
ment wishes to keep the risk of running out of stock at a point where it will
be out of stock only once every 20 weeks, or 5% of the time, then it will have
to schedule the stock replenishment when the inventory of casings on hand
drops to 66,000 units. Since the expected or average weekly usage is 50,000
units, the safety stock to be maintained is 16,000 (making a total stock of
66,000).
This example, of course, assumes a single rather arbitrary definition of
what is meant by risk or minimum acceptable level of customer service.
There are a number of ways of defining the level of service, each appro
priate to particular circumstances. One might be the total volume of
material or orders delayed; another, the number of customers delayed
(perhaps only in the case of customers with orders exceeding a certain
size level), still another the length of the delays. All of these definitions
are closely related to the "probability distribution" of sales — i.e., to the
expected pattern of sales in relation to the average.
Costs of Service Failure. It is easy enough to understand the principle
that setting a safety stock implies some kind of a management decision
or judgment with respect to the maximum sales level to be allowed for,
or the cost of service failure. But here is the rub: service failure cost,
though real, is far from explicit. It rarely, if ever, appears on the account
ing records of the company except as it is hidden in extra sales or manu
facturing costs, and it is characteristically very hard to define. What is new
in inventory control is not an accounting technique for measuring service
cost but a method of selfexamination by management of the intuitive as
sumptions it is making. The progressive company looks at what it is in
fact assuming as a servicefailure cost in order to determine whether the
assumed figure is anywhere near realistic.
For example, characteristically one hears the policy flatly stated: "Back
orders are intolerable." What needs to be done is to convert this absolute,
qualitative statement into a quantitative one of the type shown in Exhibit
II. Here we see the facts which might be displayed for the management of
a hypothetical company to help it decide on a customer service policy:
To get a 90% level of customer service (i.e., to fill 90% of the orders im
mediately), a little over three weeks' stock must be carried— an investment of
$64,000 with an annual carrying cost of $12,800.
Filling another 5% of orders immediately, thereby increasing the service
level to 95%, would mean about one week's more stock, with an extra annual
cost of $3,800.
Filling another 4% immediately (a 99% service level) would cost an extra
$7,400 per year.
GUIDES TO INVENTORY POLICY: II. PROBLEMS OF UNCERTAINTY
205
EXHIBIT II. RELATION BETWEEN SAFETY STOCKS
AND ORDER DELAY
Percentage of orders delayed
25%
20
$40,000
Annual inventory cost
$80,000 $120,000
20% annual charge for investment
— — — y— —
$16,000
$8,000
$24,000
At each point the management can decide whether the extra cost is
justified by the improved service. Thus, the chart becomes a device for
comparing policies on service and inventories for consistency and ra
tionality.
PERIODIC REORDERING
The periodic reordering system of inventory replenishment — the other
basic approach to handling uncertainty about demand — is very popular,
particularly where some type of book inventory control is employed and
where it is convenient to examine inventory stocks on a definite schedule.
The idea underlying all varieties of this system is to look at stocks at fixed
time intervals, and to vary the order amount according to the usage since
the last review.
206 INVENTORY CONTROL
The problem is that many seemingly similar ways of handling a cyclical
ordering system may have hidden traps. A typical difficulty is instability
in reordering habits and inventory levels caused by "overcompensation";
that is, by attempting to outguess the market and assuming that high or
low sales at one point, actually due to random causes, indicate an established
trend which must be anticipated. For example:
An industrial abrasives manufacturer found himself in a characteristic state
of either being out of stock or having too much stock, even though his in
ventory control procedures were, at least judging by appearances, logically
conceived. The procedures worked as follows: Each week the production
scheduling clerk examined the ledger card on each item, and each month he
placed a replenishment order on the factory based on (a) the existing finished
stock on hand in the warehouse, (b) a replenishment lead time of six weeks,
and (c) a projection for the coming twomonth period of the rate of sales
during the past twomonth period.
The manufacturer blamed the instability of his market and the perversity
of his customers for the difficulties he faced in controlling inventory, when
in fact the seemingly logical reorder rule he had developed made his business
behave in the same erratic fashion as a highly excitable and nervous driver
in busy downtown traffic. The effects of sales fluctuations tended to be multi
plied and passed on to the factory. No use was made of inventories — especially
safety stock — to absorb sales fluctuations.
The most efficient and stable reorder scheme or rule has a very simple
form:
A forecast or estimate of the amount to be used in the future is made for a
period equal to the delivery lead time plus one reorder cycle. Then an order
is placed to bring the total inventory on hand and on order up to the total
of the amount forecast for the delivery lead and cycle times, plus a standard
allowance for safety stock. Under such a scheme, the average inventory ex
pected to be on hand will be the safety balance plus onehalf the expected
usage during a reorder cycle.
Note the contrast between this scheme and that used by the abrasives
manufacturer. Here inventories are used to "decouple" production and
sales. An upward fluctuation in sales is "absorbed" at the warehouse; it is
not passed on to the plant until later (if at all). Many companies sub
scribe to this plan wholeheartedly in principle but only halfheartedly in
practice. A common tendency, for instance, is to make the forecast but
then, if sales increase, to revise it upward and transmit the increase back
to the plant. The whole value of a safety stock based on a balancing of the
costs of running out and the costs of rush orders to production is thus lost.
GUIDES TO INVENTORY POLICY: II. PROBLEMS OF UNCERTAINTY 207
Readers may recognize the application here of servo theory, the body
of concepts (including feedback, lags or reaction times, type of control,
and the notion of stability) developed originally by electrical engineers
in designing automatic or remotely controlled systems. 1 An inventory
system, though not a mechanical device, is a control system and as a con
sequence is subject to the same kinds of effects as mechanical control
systems and can be analyzed using the same basic concepts.
CHOICE OF SYSTEM
Each system of reordering inventories has it own advantages. Here are
the conditions under which the fixed order system is advantageous :
Where some type of continuous monitoring of the inventory is possible,
either because the physical stock is seen and readily checked when an item
is used or because a perpetual inventory record of some type is maintained.
Where the inventory consists of items of low unit value purchased infre
quently in large quantities compared with usage rates; or where otherwise
there is less need for tight control.
Where the stock is purchased from an outside supplier and represents a
minor part of the supplier's total output, or is otherwise obtained from a
source whose schedule is not tightly linked to the particular item or in
ventory in question; and where irregular orders for the item from the sup
plier will not cause production difficulties.
For example, the fixed order system is suitable for floor stocks at the
factory, where a large supply of inexpensive parts (e.g., nuts and bolts)
can be put out for production workers to draw on without requisitions,
and where a replenishment is purchased whenever the floor indicates the
supply on hand has hit the reorder point.
By contrast, the periodic reordering system is useful under these con
ditions :
Where tighter and more frequent control is needed because of the value of
the items.
Where a large number of items are to be ordered jointly, as in the case of
a warehouse ordering many items from one factory. (Individual items may be
shipped in smaller lots, but the freight advantages on large total shipments can
still be obtained.)
Where items representing an important portion of the supplying plant's output
are regularly reordered.
In general, since safety stocks needed vary directly with the length of
the period between orders, the periodic system is less well suited where
1 See H. J. Vassian, "Application of Discrete Variable Servo Theory to Inventory
Control," Journal of the Operations Research Society of America, August 1955, p.
272.
208 INVENTORY CONTROL
the cost of ordering and the low unit value of the item mean infrequent
large orders.
It should be noted that modifications of the simplest fixed order system
or intermediates between the fixed order system and the periodic reorder
ing system are also possible and very often useful; they can combine the
better control and cost features of each of the "pure" schemes. For ex
ample :
One type of scheme often useful — the "base stock" system — is to review
inventory stocks on a periodic basis but to replenish these stocks only when
stocks on hand and on order have fallen to or below some specified level.
When this happens, an order is placed to bring the amount on hand and
on order up to a specified maximum level.
The choice of frequency of review and the minimum and maximum in
ventory points can be determined by analysis similar to that used for the
other systems, but precautions must be taken — such as that stocks on order
must always be counted when reorder quantities are figured — in order to
avoid problems of instability and oscillation which can easily creep into
rules that are apparently sound and sensible.
Interaction Among Factors. As mathematical analysis will indicate, the
safety stock, reorder quantity, and reorder level are not entirely indepen
dent under either the fixed order or the periodic reordering system (or any
combination thereof) :
Where the order amount is fixed, the safety stock is protection against un
certainty over the replenishment time (measured by the reorder level). But
it is the size of the order amount that determines the frequency of exposure
to risk. With a given safety level, the bigger the order placed, the less frequently
will the inventory be exposed to the possibility of runout and the higher will
be the level of service.
Where inventories are reordered on a periodic time cycle, the uncertainty
against which safety stocks protect extends over the total of the reorder period
and replenishment time. But here it is the length of the reordering cycle that
determines the risk. The shorter the period and the closer together the re
orders, the less will be the chance of large inventory fluctuations and, as a
consequence, the less will be the size of safety stock required in order to
maintain a given level of service.
The interaction among the frequency of reorder, the size of reorder, and
safety stocks is often ignored as being unimportant, even in setting up
fairly sophisticated inventory control schemes (although the same com
panies readily consider the lotsize problem in relation to the other factors).
In many cases this many be justifiable for the purpose of simplifying in
ventory control, particularly methods for adjusting reorder quantities and
safety stocks to changing costs and sales. On the other hand, cases do arise
GUIDES TO INVENTORY POLICY: II. PROBLEMS OF UNCERTAINTY 209
from time to time where explicit account must be taken of such interactions
so that an efficient system may be developed.
Note, too, that the factors governing the choice of any reorder scheme
are always changing. Therefore, management should provide for routine
review of the costs of the system being used, once a year or oftener, so
that trends can be quickly identified. Also, control chart procedures, like
simple quality control methods, should be used to spot "significant" shifts
in usage rates and in the characteristics of customer demand (fluctuations,
order size, frequency of order, etc.). Schemes for checking such matters
each time a reorder point is crossed are easily incorporated in the programs
of automatic datahandling systems used for inventory control; they can
also be applied to manual systems, but less easily and hence with some
temptation to oversimplify them dangerously.
PRODUCTION SCHEDULING
Now let us turn to the important relationships between safety stocks
and production. The safety stock affects, and is affected by, production
run cycles, production "reaction times," and manufacturing capacity levels.
SETTING CYCLE LENGTHS
In production cycling problems, as in periodic reordering, the longer
the run on each product, the longer one must wait for a rerun of that
product; therefore, a larger safety stock must be maintained as protection.
Shorter, more frequent runs give greater flexibility and shorter waiting
periods between runs, and thus lower safety inventory requirements. Also,
again the interaction between factors must be taken into account. For ex
ample:
A chemical company arrived at production run cycles for a set of five
products going through the same equipment on the basis of only setup costs
and cycle inventories (e.g., lotsize inventories), ignoring the interaction be
tween cycle length and safety stocks. It found that on this basis an overall
product cycle of approximately 20 days, or one production month, appeared
optimum, allowing 4 days per product on the average. However, when the
problem was later reexamined, it was discovered that the uncertainty intro
duced by long lead times was so great that the overall product cycle could in
fact be economically cut back to less than 10 days. Doubling setup costs would
be more than offset by savings in inventory and storage costs resulting from
a reduction in the needed safety stocks.
Exhibit III illustrates the cost characteristics found to exist. The three dashed
lines show separately the annual costs of changeovers, carrying cycle inventories,
and carrying safety stocks, compared with the length of the individual produc
210
INVENTORY CONTROL
tion cycle. Adding together only the first two costs leads to the lower of the
solid lines. This is at a minimum when the production cycle is 20 days long,
indicating an apparent annual cost of $40,000. However, if all costs are in
cluded (the solid line at the top), the total annual cost on a 20day cycle is
$95,000. On this basis total costs are at a minimum when the cycle is 10
days long — only $70,000. This means a saving of $25,000 annually on the
products in question.
EXHIBIT III. INFLUENCE OF SAFETY STOCKS ON CHOICE
OF AN OPTIMUM PRODUCTION CYCLE
Annual cost
Total cost of
original cycle
$100,000
80,000 
Total cost
adjusted cycle
20,000 
_. ^ ■ — """ '. Cycle stock
5 10 15 20
Length of production cycle (days)
25
SETTING PRODUCTION LEVELS
Safety stocks give only shortterm protection against sales uncertainty.
If stocks are being replenished from production, the effectiveness of overall
control depends also on the ability to restore them in case of depletion.
If total demand varies, the ability to restore stocks depends, in turn, on
the ability of the production facilities to react to chance fluctuations. In
order to get low inventories, the process must have fast reactions properly
controlled or (equivalently) in some cases large "capacity." If reactions
are slow or limited, inventories must be large, and the inventory in effect
serves another type of protective function, namely, protection of produc
tion rate or capacity from the stresses of demand fluctuation. To illustrate
the kind of situation where this may be true:
Changes in the throughput rate of chemical processing equipment may be
slow and difficult or expensive.
GUIDES TO INVENTORY POLICY: II. PROBLEMS OF UNCERTAINTY 211
The output level of an assembly line operation may depend on the number
of stations that are manned, or the number of shifts working. Some time may
be required to change the production rate by changing the number of stations
manned at each point along the line.
The production output of a jobshop operation may be influenced by the
rate at which new workers can be hired and trained, or the cost of making
changes in the manning level by bringing in new untrained workers or laying
off people.
How fast should production operations respond to sales fluctuations,
and to what extent should these fluctuations be absorbed by means of in
ventory? The costs of warehousing and cash investment in inventory need
to be balanced against the costs of changing production rates or building
excess capacity into the production system.
The actual cost of making out schedules, which depends on the frequency
with which they are made and the degree of precision required, also should
be considered, as well as the speed of reaction of production which is
physically possible (e.g., the employee training time). When these costs
are made explicit, management may find itself having to balance conflicting
objectives. To illustrate:
A metal fabricator making a wide line of products to order attempted to
provide immediate service to customers. He found that on the average his
departments needed a substantial excess of labor over the normal requirements
of the jobs flowing through, and this excess was essentially idle time. On the
other hand, when he attempted to cut the excess too thin, backlogs began
to build up. He had to weigh his desire to get the lead time down against
the costs of excess unused labor.
Ordinarily we want to avoid passing back the full periodtoperiod sales
fluctuation by making corresponding changes in the size of orders placed
on production because it is uneconomical. What we can do instead is to:
1 . Set the production level in each period equal to anticipated needs over the
lead time plus the scheduling period not already scheduled, plus or minus some
fraction of the difference between desired and actual inventory on hand.
2. Alternatively, change the existing production level or rate by some frac
tion of the difference between the existing rate and the rate suggested by the
simple reorder rule (i.e., that an order be placed in each period equal to the
anticipated requirements over the lead time plus the scheduling period, plus
or minus the difference between desired and actual inventory on hand and on
order).
Each of these alternatives is useful in certain types of plants, depending
on whether the cost of production fluctuations comes primarily from, say,
overtime and undertime (work guarantee) costs or from hiring, training,
212 INVENTORY CONTROL
and layoff costs. Each in appropriate circumstances will lead to smoother
production, at the expense of extra inventory to maintain the desired level
of service.
When the different costs involved are identified and measured, mathe
matical techniques can be used to show the effect that varying the numbers
in the rule (in particular, the size of the fraction used) has on inventory
and production expense and to arrive at an economical balance between
the needs of marketing and manufacturing. These two rules are expressions
of servo theory, like that referred to earlier in connection with inventory.
Here it may be worthwhile to see in some working detail how the theory
can be applied mathematically:
The first rule can be stated as follows:
P i is the amount scheduled for production in period i, F i is the forecast re
quirements for period i, I is the desired inventory, l { is the actual opening
inventory on hand in period i, and k is the response number which indicates
what fraction of the inventory error or production rate departure is to be
accounted for each period.
The fluctuations in inventory resulting from a choice of k in the first rule
can be expressed as a function of the fluctuations in sales about the forecast,
as follows (if fluctuations from month to month are not correlated) :
T(2k  k 2 ) + 1
where 07 is the standard deviation of inventory levels, and a F is the standard
deviation of actual sales about forecast sales each period. Similarly, the produc
tion rate variations resulting from any choice of k can be expressed as:
dp — / <TF
2 k
The influence of the choice of a response number, k, on the standard devia
tion of inventories and on the standard deviation of production rates under the
first type of rule is shown in Exhibit IV. Frequently the costs of production
fluctuations are more or less directly proportional to the standard deviation of
fluctuations in the production rate, a measure of the amount of change in pro
duction level which can be expected to occur. On the other hand, the normal
inventory level, the average level expected, must be set large enough so that
even with expected inventory fluctuations, service failures will not occur exces
sively. This means that the larger the standard deviation in inventory levels, the
larger must be the normal level, generally in proportion. Therefore, one can
GUIDES TO INVENTORY POLICY: II. PROBLEMS OF UNCERTAINTY
213
"buy" production flexibility with larger inventories, and vice versa, with the
particular costs in the process concerned determining the economical balance. 2
The second rule can be worked through similarly. Here P* is the changed
amount scheduled for production, and the rule can be stated as follows:
where
P*= PLi + k{P i  ??_!) ; k ^ 1 = (1  K) PJL X + kP t
Pi= 5 F i+k  I P*_ h + U h)
fc=0 fc=l
EXHIBIT IV. EFFECT OF RESPONSE NUMBER K ON VARIATIONS
IN INVENTORY AND PRODUCTION RATE
1.0
.5
t 1 r
.4 .6 .8
Response number k
SETTING CAPACITY LEVELS
In some cases — particularly where output cannot be stocked easily —
the problem of controlling the production level is not so much one of
adjusting the level to respond to fluctuations in demand, as of setting the
capacity of the plant or operation at a high enough level to permit demand
fluctuations to be absorbed without excessive delay. If the capacity is set
2 See H. J. Vassian, op. cit. See also Charles C. Holt, Franco Modigliani, and
Herbert A. Simon, "A Linear Decision Rule for Production and Employment Schedul
ing," Management Science, October 1955, for another approach to this problem under
different cost conditions.
214 INVENTORY CONTROL
equal only to the desired average rate, fluctuations in demand about this
desired rate must either be absorbed by inventories or by orders piling
up in a backlog. To illustrate:
The telephone companies have recognized for many years that telephone
exchanges must be built with greater capacity than is required to handle the
average load, in order to keep lines of waiting subscribers within reasonable
levels.
Pileups often occur around the checkout booths of cafeterias or the ticket
windows in railroad stations. Customers are eventually taken care of, but
capacity is so close to average requirements, in some cases, that long waiting
lines can be built up as a result of customers arriving at random in small
bunches.
The problem of specifying the number of workmen to tend semiautomatic
machinery or the capacity of docks to service freighters is complicated by the
fact that the units require service more or less at random, so that again there
can easily develop an accumulation of units awaiting service if personnel are
not immediately available.
A theory of such processes is growing; it is known as waitingline
theory. This is really a branch of probability theory, and is itself a whole
body of mathematical techniques and explicit concepts providing a mathe
matical framework within which waitingline and similar problems can be
studied. 3
Some examples of applications in production scheduling are: flow of
orders through departments in a job shop; flow of items through the stages
in an assembly line; clercial processing of orders for manufacture or ship
ping; filling orders in a warehouse or stockroom; and setting up shipping
or berth facilities to handle trucks or other transport units. In each case,
fairly wellfixed crews or facilities have to be set up for handling fluctuating
orders or items quickly, avoiding delays in service. A balance between the
cost of extra personnel or facilities and delays in taking care of demand is
needed.
In applying waitingline theory to such problems, the flow of orders or
demand for goods can be considered as a demand for service, analogous to
subscriber cost in a telephone exchange. Orders are handled by one or
more processing stations, analogous to telephone trunk lines. When the
order or unit is produced, the processing station is free to take on the next
order in line, as when a call is completed through the exchange. For
example :
3 A technical discussion of waitingline theory and related applications can be
found in W. Feller, An Introduction to Probability Theory and Its Applications (New
York, John Wiley & Sons, Inc., 1950), Chapter 17.
GUIDES TO INVENTORY POLICY: II. PROBLEMS OF UNCERTAINTY 215
A wholesale merchandise house planned its orderhandling and orderfilling
activities in advance of peak sales. The company, selling consumer merchandise
to a large group of retail dealers, had grown rapidly and in midsummer had
looked forward to serious congestion, delayed orders, and lost customers when
the Christmas peak hit. An analysis based on waitingline theory outlined staff
and space requirements to meet the forecast load, showed what jobs were
the worst potential bottlenecks, and revealed, incidentally, how the normally
inefficient practice of assigning two persons to "pick" one order could in this
case help avoid tieups and save space during the critical sales peak.
STAGES OF CONTROL
The choice and use of appropriate techniques for inventory control is
not a simple matter. It takes a good deal of research into sales and product
characteristics, plus skill in sensing which of many possible approaches are
likely to be fruitful.
To describe these techniques, I shall take a case illustration. This case
is drawn from a great deal of business experience, but in order to keep
the detail and arithmetic within manageable proportions without distorting
the essential points, I have simplified and combined everything into one
fictional situation.
Any of the stages of the company's progress toward more efficient in
ventory management — from the original to the final — might be found to
exist in the inventory control practices of a number of sizable companies
with reputations for progressive and efficient management. These stages of
advancement in the refinement of inventory control should not be used to
compare the inventory system of one company or division with that of
another, for the reasons just mentioned; but they may prove helpful to
management in answering the questions, "Where are we now?" and "What
could we do better?"
Briefly, the case situation is as follows:
One division of the Hibernian Bay Company makes and sells a small
machine part. Sales run slightly over 5,000 units annually, and the price
is $100 apiece. Customers are supplied from four branch stock points
scattered about the country, which in turn are supplied by the factory
warehouse. The machining and assembly operations are conducted in a
small plant, employing largely semiskilled female help. The level of pro
duction can be changed fairly rapidly but at the cost of training or re
training workers, personnel office expenses, and increased inspection and
quality problems. The division management has almost complete autonomy
over its operations, although its profit records are closely scrutinized at
headquarters in Chicago.
216 INVENTORY CONTROL
Originally the factory and branch warehouse stocking practices were
haphazard and unsatisfactory. In total, nearly four months' stock was
carried in branches, in the factory warehouse, or in incompleted produc
tion orders. A stock clerk in each branch who watched inventories and
placed reorders on the factory warehouse was under pressure to be sure
that stocks were adequate to fill customer orders. The factory warehouse
reorder clerk in turn watched factory stocks and placed production orders.
Production runs or batches were each put through the plant as a unit.
Fluctuations in production, even with apparently sizable stocks on hand,
caused the management deep concern.
SERVICE IMPROVED
The management decided to try to improve inventory practices and
appointed a research team to study the problem. The team suggested using
"economical order quantities" for branch orders on the factory warehouse
and warehouse orders on production, as a basis for better control. The
steps followed were:
The research team suggested that the formula for determining the
economical order quantity was x =± \/2As/I, where A = fixed cost con
nected with an order (setup of machines, writing order, checking receipts,
etc.), i = annual cost of carrying a unit in inventory, s = annual move
ment, and x = "economical order quantity."
The team found that each branch sold an average of 25 units a week,
or 1,300 per year; that the cost of a branch's placing and receiving an order
was $19 ($6 in clerical costs at the branch and factory, $13 in costs of
packing and shipping good, receiving, and stocking) ; that annual inventory
carrying costs in the branches were $5 per unit, based on a desired 10%
return on incremental inventory investment. The reorder quantity for each
branch was computed as \/2 • $19 • l,300/$5 = 100 units reorder quantity.
A system was set up where each branch ordered in quantities of 100, on
the average, every four weeks. On this basis, without further action, each
branch would have had an average inventory of onehalf a reorder quantity,
or 50 units. (The books would show 75 units, since stock in transit from
factory warehouse to branch was also charged to the branch, and with
average transit time of one week this would average 25 units.)
The next step was to provide for enough to be on hand when a reorder
was placed to last until the order was received. While the average transit
time was one week, experience showed that delays at the factory might
mean an order would not be received at the branch for two weeks. So sales
for two weeks had to be covered.
GUIDES TO INVENTORY POLICY: II. PROBLEMS OF UNCERTAINTY
217
Statistical analysis showed that sales in any one branch over two weeks
could easily fluctuate from 38 units to 62 units and could conceivably go as
high as 6570. The management decided that a 1 % chance of a branch
running out of stock before getting an order would be adequate.
Calculations then indicated that the maximum reasonable twoweek
demand to provide for would be 67. (The statistical basis was that sales
fluctuate about the average at random; that fluctuations in the various
branches are independent of one another; and that the standard deviation
is \fsi where s = sales rate, and t = length of individual time period.)
The branches therefore were instructed to order 100 units whenever the
stock on hand and on order was 67 or less. This gave an inventory in each
branch made up on the average as follows:
Safety stock
Arl
(order point, 67, less normal
week's usage, 25)
Order cycle stock
50
(one half 100unit order)
In transit
25
(one week's sales)
Total
117
or 4.7 weeks' sales
The resulting behavior of the reorder system is shown in Exhibit V —
both as it would be presumed in theory and as. it actually turned out.
Although the actual performance was much less regular than presumed,
the two compare fairly well — testimony to the soundness of the procedure.
EXHIBIT V. ECONOMICAL REORDER SYSTEM OF A
BRANCH WAREHOUSE
Inventory
A. Presumed Operation
B. Actual Operation
120
100
N. Order
80
\^ placed
Order placed
60
. Reorder ^k
Point \
40"
Safely stock
20
T ' I
I 1   I 
2 3 4 5
Time (weeks)
Inventory
218 INVENTORY CONTROL
APPLICATION AT THE FACTORY
At the factory warehouse end, the "economical order quantity" scheme
worked as follows:
The cost of holding a unit in inventory was $3.50 per year (at 10% return
on investment); the cost of placing an order and setting up equipment for
each order was $13.50; and, of course, a total of 5,200 units was made each
year. These indicated that each production order should be for V2 • $13.50
•5,200/$3.50 = 200 units.
Factory processing time was two weeks; it would take two weeks for each
order to reach the warehouse. The warehouse would need to place its replenish
ment order on the factory when it had enough on hand or on order to fill
maximum reasonable demand during the next two weeks.
On the average, the factory warehouse would receive one order a week from
the branches (one every four weeks from each of four branches) under the
new branch reorder system. In fact, because of the fluctuations in branch sales
described before, it was found that orders on the factory warehouse fluctuated
substantially in any twoweek period (see Exhibit VI).
EXHIBIT VI. FLUCTUATIONS OF ORDERS ON FACTORY
WAREHOUSE
Number of Number of items Percentage
branch orders ordered of weeks
A. Weekly Periods
37%
1 100 37
2 200 18
3 300 6
4+ 400+ 2
B. Biweekly Periods
13%
1 100 27
2 200 27
3 300 18
4 400 9
5 500 4
6 600 1
7+ 700+ 1
It was agreed that to give branches service adequate to maintain their
own service, stocks at the factory warehouses would have to be high
enough to fill demand 99% of the time, i.e., a replenishment order would
have to be placed when 600 units were on hand. This meant a safety stock
of 600 units minus 200 (normal usage), or 400 units. Cycle stock averaged
GUIDES TO INVENTORY POLICY: II. PROBLEMS OF UNCERTAINTY 219
half a run, or 100 units, and stock in process an additional half run, or
100 units. Total factory stock, then, was:
Cycle stock 100 units
Stock in process 100 units
Safety stock 400 units
Total 600 units
Exhibit VII gives a picture of the apparent costs of the "economical
order" system. The stock of 1,068 units equaled less than 11 weeks' sales,
a fairly substantial reduction, and the management felt that it had a better
control, since clerical procedures were set up to adapt readily to any
changes in inventory charges (currently 10% per year) or service level
requirements the management might choose to make.
EXHIBIT VII. COSTS OF REORDER SYSTEM
Number
Cost each
Annual cost
Inventory
Factory
600 units
$3.50/year
$2,100
4 branches
468 units
$5.00/year
2,340
Reorder cost
Branch
52/year
$19.00
990
Factory
26/year
$13.50
350
Total
$5,780
PRODUCTION STABILIZED
But the factory still had problems. On the average, the warehouse would
place one production order every two weeks, but experience showed that
in 60% of the weeks no orders were placed, in 30% one order, and in
10% two, three, or more orders were placed.
Factory snarls due to these fluctuations occasionally caused the factory
to miss deadlines. These in turn led on occasion to warehouse delays in
filling branch orders, and forced the branches to hold to the twoweek
delivery time even though actual transit time was only one week. An
analysis revealed the following:
Factory fluctuations were very costly. A statistical regression of costs against
operating levels and changes showed that annual production costs were affected
more by the average size of changes in level than by the frequency of change;
a few large changes in operating level were much more costly than many small
changes.
Under the "economical reorder quantity" system, production fluctuations
were no larger than before, but the average change up or down actually equaled
220 INVENTORY CONTROL
80% of the average production level. This was estimated to cost $11,500 an
nually, bringing the total cost of the system, including costs of holding inven
tories, placing orders, and changing production rates, to $17,280 per year.
This led to the suggestion that the company try a new scheme so that
orders on the factory warehouse and the factory would be more regular.
A system with a fixed reorder cycle or period was devised, under which
branch warehouses would place orders at fixed intervals, the order being
for the amount sold in the period just ended. The factory warehouse would
ship the replenishment supply, order an equivalent amount from the factory,
and receive the order within two weeks or by the beginning of the next
review period, whichever was longer.
Under this scheme, each branch warehouse would need to keep its
stock on hand or on order sufficient to fill maximum reasonable demand
during one review period plus delivery time (tentatively taken as two
weeks) on the basis of the reorder rule described previously in this article.
The question to be determined was: How long should the review period,
that is, the time between reorders, be? Exhibit VIII summarizes inventories
and costs for reorder intervals ranging from one to six weeks, based on
the following facts and figures:
( 1 ) Branch safety stock was determined from a study of branch sales fluctua
tions, to allow for maximum reasonable demand over the reorder interval plus
the twoweek delivery period.
"Maximum reasonable demand" was defined to allow a 0.25% risk of being
out of stock in any one week (equal to the 1 % risk on the average fourweek
interval under the "economical reorder quantity" system described previously).
(2) Branch cycle stock would average onehalf of an average shipment.
Under this system, the average shipment to a branch each period would equal
the average sales by the branch in one period (25 units X number of weeks).
(3) Transit stock equaled one week's sales.
(4) Branch inventory carrying cost was $5 per unit per year.
(5) Branch ordering costs equaled $19 per order, with one order per period.
A oneweek period would mean 52 orders per year; a twoweek period, 26
orders per year; etc.
(6) Factory safety stock was set to allow a 1% risk that the warehouse
would be unable to replenish all branch shipments immediately.
(7) Factory cycle stock in process or in the warehouse would be approxi
mately equal to onehalf the sales in any one period.
(8) Factory inventory carrying cost was $3.50 per unit per year.
(9) Factory ordering costs equaled $13.50 per order (see 5 above).
(10) Production change costs were proportional to the periodtoperiod
changes in production level, equal under this system to periodtoperiod changes
in branch sales.
GUIDES TO INVENTORY POLICY: II. PROBLEMS OF UNCERTAINTY
221
EXHIBIT VIII. SUMMARY OF REORDER PERIOD
COST COMPARISONS
Length of period {weeks)
4
Branch warehouse
Safety stock
Cycle stock
Transit stock
Total units of stock
Annual inventory cost
Ordering cost
Total cost each branch
Total cost four branches
Factory warehouse
Safety stock
Cycle stock
Total units of stock
Annual inventory cost
Ordering cost
Total cost factory
Production change costs
Total system costs
24.0
12.5
25.0
61.5
$ 310
990
$1,300
$5,200
26.0
25.0
25.0
27.0
37.5
25.0
28.0
50.0
25.0
76.0
89.5
103.0
30.0
62.5
25.0
117.5
$ 380 $ 450 $ 515 $ 590
495 330 250 195
$ 875 $ 780 $ 765 $ 785
$3,500 $3,120 $3,060 $3,140
31.0
75.0
25.0
131.0
$ 650
165
$ 815
$3,260
33
50
33
100
41
150
191
$ 670
235
47
200
52
250
58
300
83
$ 290
700
133
$ 465
350
247
$ 865
175
302
$1,060
140
358
$1,250
120
$ 990
$1,600
$ 815
$2,250
$ 905
$2,760
$1,040
$3,180
$1,200
$3,560
$1,370
$3,900
$7,790 $6,565 $6,785 $7,280 $7,900 $8,530
The figures show that a twoweek reorder interval would be most
economical for the company as a whole, and this was chosen. Costs were
estimated to be $6,600, compared with $17,300 under the "economical
reorder quantity" system. While the new system cut total inventories by
nearly 70%, most of the gain came from smoother production operations.
Further economies became apparent when the system was in operation:
( 1 ) The reduction in production fluctuations made it possible to meet
production deadlines regularly, cutting the effective lead time in deliveries to
branches and thereby permitting modest reductions in branch safety stocks.
(2) The inventory system was found well suited to "open" production orders.
Instead of issuing a new order with each run, the moderate fluctuations made
it possible to replace production orders with simplified "adjusting memos" and
at the same time to eliminate much of the machine setups.
"BASE STOCK" SYSTEM
The success with the periodic reordering system encouraged the com
pany to go further and try the "base stock" system referred to earlier. Under
this system, the branch warehouses would report sales periodically. The
222 INVENTORY CONTROL
factory would consolidate these and put an equivalent amount into produc
tion. Stocks at any branch would be replenished whenever reported sales
totaled an economical shipping quantity.
Two possible advantages of this system compared to the fixed period
scheme were: (1) Branches might be able to justify weekly sales re
ports, reducing production fluctuations and safety stock needs still further.
(2) It might be possible to make less frequent shipments from factory to
branches and make further savings. The following questions had to be
decided:
How frequently should branches report sales? As noted earlier, cost studies
showed that of the $19 total cost of ordering and receiving goods $6 represented
clerical costs in placing and recording the order. Here is a summary of the
costs affected by the choice of reporting interval:
Reporting interval
One
week
Two
weeks
Number
Cost Numb
er
Cost
Branch safety stock 100 $ 500 108 $ 540
Production changes 1,600 2,250
Branch clerical costs 4 X 52 1,250 4 X 26 625
Total $3,350 $3,415
Thus, there appeared to be some advantage to reporting sales weekly from
branches to the factory.
How big should replenishment shipments be? Exhibit IX summarizes the
system costs related to the size of shipment from factory to branch. Each line
shows the total of the cost indicated plus those represented by the line below.
The total system cost (top line) is lowest at 82; that point is therefore the
optimum shipping quantity from factory to branch warehouse. The same
answer can be obtained from the formula given before, V2 • $13 • l,300/$5
= 82.
The base stock system therefore was set up with weekly reporting and
replenishment shipments of 82 units to branches. The total cost of the
base stock system was $5,200 compared with $6,600 under the previous
system.
STABILIZED FURTHER
The company, cheered by its successes, decided to see if even further
improvements might be obtained by cutting down further on production
fluctuations. As it was, the production level under the base stock system
was being adjusted each week to account for the full excess of deficiency
GUIDES TO INVENTORY POLICY: II. PROBLEMS OF UNCERTAINTY
223
in inventory due to sales fluctuations. It was proposed that production be
adjusted to take up only a fraction of the difference between actual and
desired stocks, with added inventories used to make up the difference.
EXHIBIT IX. OPTIMUM SHIPPING QUANTITY FROM FACTORY TO
BRANCH WAREHOUSE UNDER BASE STOCK SYSTEM
COST
$7,000
$6,000
\ Total system cost
> Shipping cost
$5,000
$4,000
• ^*»"" \ Inventory cost related
^W""" r to order quantity
$3,000
\ > Reporting cost
$2,000
$1,000
> Production changes cost
 Safety stock
25 50 75 100 125 150
Size of shipment (units)
The two costs that would be affected are costs of changing production
and costs of holding inventories, in particular safety stocks. These are
affected by the fraction of the inventory departure that is made up each
week by adjusting production.
The study showed that the cost would be minimized with the rate of
response set equal to 0.125. (This compared with a response rate of 1.0
under the base stock system.) The additional savings of $970 brought the
annual cost of the system down to $4,200.
SUMMARY
The results of all the changes made by the division management were
substantial :
(1) A major reduction in stocks — They had been cut 35% from what
they were even with the "economical reorder quantity" system.
(2) A substantial reduction in production fluctuations — The problems of
the case are common even among the bestrun businesses and can be solved
in much the same way with much the same results. Of course, a large part
224 INVENTORY CONTROL
of the effort and expense that were necessary in this stepbystep, evolu
tionary approach could be saved. Technical methods are available for
analyzing and measuring the performance of alternate systems so that
management can proceed directly to the ultimate system that is most de
sirable; management does not have to feel its way. Let me emphasize
again, however, that no one kind of system should be considered "the goal."
The efficiency of any given inventory control plan depends too much on
the demand and cost characteristics of the business.
In the discussion thus far, several large questions remain unanswered.
What happens when the business is subject to seasonal sales? What more
can be done than to insure that desired levels of service are maintained
while cutting inventory and production costs? Where do forecasting and
scheduling fit into the picture? I shall discuss these questions in the next
and final article in this series.
Chapter 11
COMBINED PROBLEMS OF
INVENTORY AND PRODUCTION
CONTROL
XIII.
Guides to Inventory Policy
III. Anticipating Future Needs*
John F. Magee
Businessmen are prone to view inventories with distaste, as an apparently
necessary drain on resources, something that no one has been able to
eliminate but hardly a "productive" asset like a new machine or tool.
In fact, however, inventories are as productive of earnings as other types
of capital investment. They serve as the lubrication and springing for a
productiondistribution system, keeping it from burning out or breaking
down under external shocks. They help to absorb the effects of errors in
forecasting demand, to permit more effective use of facilities and staff
in the face of fluctuations in sales, and to isolate one part of the system from
the next in order to permit each to work more effectively.
* From the Harvard Business Review, Vol. 34, No. 3 (1956), 5770. Reprinted by
permission of the Harvard Business Review.
225
226 INVENTORY CONTROL
In this article let us look at the function of a third type of inventory,
one which is of particular importance in longrange planning: anticipation
stocks. This type of inventory is most commonly needed where sales are
highly seasonal, and where either one or the other of these problems
occurs :
1. The "crash" or shortpeak season problem which arises, for example, in
the toy industry before Christmas or in certain fashion clothing lines at
various times during the year.
2. The more conventional seasonal problem arising in industries where
sales show a pronounced seasonal swing, with the peak season often extend
ing over several weeks or months, as in the case of automobiles, many kinds
of building materials, certain cosmetics, some types of home appliances, agri
cultural supplies, and furniture.
Stocks built up to buffer production against seasonal fluctuations in
sales are not the only form of anticipation stocks. Anticipation stocks may
also be carried, for example, to meet a planned intensive sales campaign or
to carry sales over a plant vacation or maintenance shutdown. However,
the questions and methods of attack which apply to seasonally fluctuating
sales also illustrate approaches to control of other types of anticipation
stocks; I shall therefore use the former as a basis of discussion in this
article.
THE "CRASH" PROBLEM
In the "crash" type of problem, management must balance the risks of
not having enough stocks to fill demand and thus losing profit, or of being
forced to go to extraordinary measures to buy or produce to fill demand,
against the risks of having too much on hand and consequently incurring
sizable writeoff and obsolescence loss or storage expense until the next
selling season.
The question boils down to how much stock to have on hand when
the main selling season opens. The objective basically is to have enough
on hand so that the company can expect, on the average, to break even
on the last unit produced; that is, to carry enough so that on the last unit
the expected risk of loss due to inability to fill demand equals the expected
cost of carrying the unit through to the next season.
METHOD OF APPROACH
In principle, the solution to the "crash" problem is quite simple. The
classic "newsboy" case is as good an illustration as any:
A newsboy has, on the average 10 customers a night who are willing to
buy papers costing 5<t each. The newsboy makes a profit of 3^ on each
paper he sells, and loses H on each paper he takes out but fails to sell. Let
GUIDES TO INVENTORY POLICY: III. ANTICIPATING FUTURE NEEDS 227
us suppose he has kept records, and that 40% of the time he can sell at least
10 papers and 20% of the time he can sell at least 12 papers.
If the newsboy does not know how many papers he will actually sell in
any given day but every day takes out 10 papers, he has a 40% chance of
selling all the papers and making 34 each, and a 60% chance of not selling
all papers and losing \4 on each not sold. He can expect the tenth paper
to produce, on the average over time, a profit of 0.64 (34 X 40% — \4 X60% ).
On the other hand, if he takes 1 2 papers every night, he can expect the twelfth
paper to produce, on the average over time, a loss of 0.24 (3 X 20% — \4 X
80%).
It would not, therefore, be worth his while to take out 12 papers. As a
matter of fact, he would probably make the greatest total profit by taking
1 1 regularly, since he could expect, on the overage over time, to do slightly
better than break even on the eleventh paper (34 X 30% — 14 X 70% , or 0.2<0 .
The newsboy problem is, after all, not so different from many business
problems. Certainly from the newsboy's point of view the papers he buys
which he may not sell represent a lot of money and a sizable risk of his
capital. Indeed, perhaps the most important difference between the news
boy and businessmen in other situations is that the newsboy has to make
this decision very frequently and therefore has more of a chance to build
up a lot of experience on which to base intuitive judgments — that is, less
need for careful calculation or formal statistical methods to wring out of
past experience the information which is of value.
REACHING A SOLUTION
Suppose, for example, you are selling cosmetics and you want to make
up a special Christmas package in a holiday wrapping containing three
normally separate items at a combined price. You have tried a number of
deals of this type in the past, and on the whole they have been highly
profitable. However, individually they have been unpredictable; some have
been very successful, and some that seemed excellent on paper turned out
to be failures.
Your market research manager makes a volume prediction each time;
on the average, his estimates come fairly close, but rarely on the nose.
About half the time they are too high and half the time too low. In fact,
25% of the time your experience shows his estimate to be 20% or more
on the high side, and just as frequently he misses as badly in the other
direction. About 10% of the time he is as much as 40% off in each
direction, and occasionally he really misses and actual sales are 75% or
more off from the estimate. You are doing everything you can to improve
these estimates, but in the meantime you have to decided how much to
make up for your Christmas deal.
228 INVENTORY CONTROL
EXHIBIT I. GENERALIZED MATHEMATICAL EXPRESSION OF
APPROACH TO "CRASH" PROBLEM
Let:
V = volume of demand
f(V) — the probability density function of demand (i.e., distribution of demand
during one period)
oo
$f{V)dV = the likelihood of selling an amount Y or more during a season
Y
n — the variable cost of making and holding a unit of stock in inventory dur
ing the selling period, including the capital charge for inventory invest
ment, etc.
m = the profit per unit sold
L = the cost per unit of not filling an order (loss of good will), over and
above the loss of profit
P = the cost of carrying a unit of inventory if unsold by the end of the period
K = the size of the inventory on hand at the beginning of the season
Then the profit earned during the replenishment cycle is given by:
pmV P(K  V)  nK; V^K
= mK L(V  K) — nK; V > K
and the expected profit earned during the replenishment cycle, E(p), is given by:
E(p) = m / Vf(V)dV nK + mK*$ \{V)dV 
K
l] (V  K)f{V)dV  P J (K  V)f(V)dV
K
Again, differentiating the expected profit with respect to the inventory on hand at
the beginning of the season, K, yields:
dE(p) K
K ZL   n + (m + L)  (M + L + P) J f(V)dV
dK
The maximum profit will be earned when dE(p)/dK = 0; that is, when
m 4 L + P
Cost estimates indicate that if a package is not sold, the items can be
repackaged at an extra cost of about $1 per package. If demand exceeds
the original run, the extra cost of a special rerun plus emergency ship
ments to field stocks is estimated to be $1.75 per package. Following
reasoning like that in Exhibit I (simply a generalized expression in mathe
matical terms of the solution to the newsboy case), you or your operations
research analyst concludes that you should plan initially to have enough
stock so that the chance that demand, as it materializes, will be covered
by the initial run equals the ratio of the special makeup cost to the total
of (a) special makeup cost plus (b) repackaging loss on unsold items.
In other words, you want to make enough so that the chance that total
sales will be covered by the initial run equals $1.75 r ($1.00 + $1.75)
GUIDES TO INVENTORY POLICY: III. ANTICIPATING FUTURE NEEDS 229
or 64%. With your past experience on forecasting success, this means
about a 10% overstock; that is, your initial run should exceed your
estimated needs by about 10%.
This will not eliminate all the difficulties by any means. There is nearly
a 40% chance you will have to make some additional highcost stock, and
there is still a good chance you will have unsold goods on hand after the
holiday. However, this initial decision is about the best you can do with
present forecasting and manufacturing methods to get the right balance
between the two risks and thus minimize the over all cost.
In problems like those noted above, the costs may, superficially at least,
look different, and the mathematical details of formulating an approach
to the problem and arriving at an answer may differ, but the basic elements
are the same — balancing the cost and lost profit opportunities of demand
exceeding available stock against the costs and losses of having available
unused stock or capacity.
DEVELOPING APPROACH
Sometimes from the scanty experience gained in earlyseason selling
enough information can be developed so that estimates of total season
sales and resulting production plans can be adjusted. As more and better
information becomes available, mathematical methods can be used to alter
the "strategy" for the season slowly, according to predetermined rules.
Such a "developing" approach to inventory problems rests on the basic
premises that one does not know the future, that there is therefore no
need to plan into it very far in great detail, and that a good strategy for
the present is one which puts you in a position to make a good choice the
next time you have a chance, whatever actual experience may develop in
the meantime. Applications of this general line of approach to problems
are beginning to be made in the planning of heatingoil production,
seasonal clothing production, and other seasonal, erratic demand problems.
SEASONAL SWINGS
In many industries, the basic yearly pattern of seasonal sales may be
quite predictable, and the overall volume can be reasonably well esti
mated. There may be only a small error of a few percentage points in
estimating either the total volume or the size of the peak. In situations of
this sort there are three problems:
1 . Adjusting the forecast of expected sales to allow for safety stocks so as
to protect against forecast errors. (Examples of an original and an adjusted
"maximum" sales forecast are shown in Exhibit II. The latter is the original
cumulative forecast increased by the safety stock allowance.)
230
INVENTORY CONTROL
2. Laying out a production pattern or plan to meet the forecast. (The
difference between forecast and production plan will result in a planned in
ventory as illustrated in Exhibit II. The total costs of inventory and produc
tion depend on the form of the production curve, and characteristically the
object is to choose this curve or production plan to minimize the expected total
of these costs.)
3. Controlling or adjusting the production plan to keep it aligned with the
sales forecast, as actual sales experience modifies the forecast and/ or results
in depleted or excessive inventory as compared with the plan.
MEETING FORECAST ERRORS
The answer to the first problem depends somewhat on that for the
third, as the discussion on production control rules in the second article
in this series may suggest. In general, however, it is fair to say that in most
businesses the risks and costs of back orders so outweigh inventory cost
that substantial protection in the form of safety stocks is justified. These
EXHIBIT II. ILLUSTRATIVE SALES FORECAST AND
PRODUCTION AND INVENTORY PLAN
QUANTITY IN UNITS
110,000 h
100,000 
90,000 
80,000 
70,000 
60,000 
50,000
CUMULATIVE
PRODUCTION PLAN
MAXIMUM
CUMULATIVE
SALES FORECAST
PLANNED
40,000  INVENTORY
30,000
20,000 k
10,000
SAFETY STOCK
CUMULATIVE
SALES FORECAST
f*\" I 1 1 1 1 1 1 1 1 h
JAN FEB MAR APR MAYJUNE JULY AUG SEPT OCT NOV DEC
TIME OF YEAR
GUIDES TO INVENTORY POLICY: III. ANTICIPATING FUTURE NEEDS
231
safety stocks must be large enough so that stocks can be restored after a
sudden unexpected sales spurt by a smooth and moderate adjustment in
production rate. The production response rules described in the previous
article, which take into account the nature of forecast errors, inventory
costs, and service requirements, are one way of determining what is
"large enough."
Another very similar approach is to begin with a forecast of maximum
expected demand, or maximum demand the company is prepared to plan
for. The longrange production plan is made out to meet this directly.
Then production is adjusted downward from plan as excess inventories
accumulate because of actual sales falling below the maximum plan. (More
will be said later about this problem of production control.)
PLANNING PRODUCTION
Once the adjusted sales forecast or forecast plus safety stock has been
obtained, the task is to plan the production rate or draw in the production
curve shown in Exhibit II. The problem is to find a curve or mathematical
function that will minimize the total of production and inventory costs.
In theory, this sounds like a straightforward mathematical problem often
encountered in physics. In practice, the job is not so easy, but a number
of techniques have been found useful.
Graphical Techniques. Where the problem of planning production
against forecasted seasonal sales is not made too complicated by a variety
of items, processes, and stages, simple graphical or arithmetic techniques
can often be useful. For example:
Suppose a company has a forecast at the beginning of the year which calls
for requirements as outlined in Exhibit III. The first column shows expected
sales month by month; the second column shows accumulated expected sales;
EXHIBIT III. FORECAST OF SALES AND SAFETY STOCKS NEEDED
(In units)
Cumulative
Cumulative
Expected
sales
Safety
total re
Month
sales
forecast
reserve
quirements*
January
6,000
6,000
3,000
5,500
February
4,000
10,000
2,500
9,000
March
3,000
13,000
2,100
11,600
April
4,000
17,000
2,500
16,000
May
6,000
23,000
3,000
22,500
June
9,000
32,000
3,500
32,000
July
11,000
43,000
4,000
43,500
August
12,000
55,000
4,200
55,700
September
13,000
68,000
4,400
68,900
October
12,000
80,000
4,200
80,700
November
11,000
91,000
4,000
91,500
December
9,000
100,000
3,500
100,000
Less opening stock of 3,500
232
INVENTORY CONTROL
the third column shows a safety reserve to cushion the company against fore
cast errors, allowing time for smooth adjustment (the basis for this reserve
will be discussed further below) ; and the last column shows the total amount
that must be produced by the end of each month, allowing for an opening
stock of 3,500 units.
The cumulative forecast and cumulative requirements, including opening
stock, are shown in Exhibit IV. The company could produce at an average
annual rate of 100,000 units, or 8,333 units per month — the production plan
shown as a straight line in the exhibit. This plan would produce just enough
inventory at the endofyear peak to meet requirements. The monthend in
ventories (equal to the difference between the production plan and the cumu
lative sales forecast) are shown in Exhibit V. They average 12,800 units, of
which 3,400 are accounted for as safety stock, leaving an average seasonal
anticipation stock of 9,400 units. If the annual inventory carrying cost were
$45 per unit, the seasonal anticipation stocks would be costing about $425,000
per year.
EXHIBIT IV. ACCUMULATIVE SALES FORECAST AND
ALTERNATE PRODUCTION PLAN
QUANTITY IN UNITS
110,000 
100,000
90,000
80,000
70,000
60,000
50,000
40,000
30,000
20,000
10,000
UNIFORM PRODUCTION
PLAN
ALTERNATE PRODUCTION/ //;
PLAN / '/
//
,/.• CUMULATIVE SALES
 FORECAST
,'y/ CUMULATIVE PRODUCTION
REQUIREMENTS
■I 1— I 1 1 1 1 1 1
JAN FEB MAR APR MAY JUNE JU LY AUG SEPT OCT NOV DEC
TIME OF YEAR
GUIDES TO INVENTORY POLICY: III. ANTICIPATING FUTURE NEEDS 233
Various alternatives might be tried to reduce this cost. For example, opera
tions might be run during the low months of the year at the rate of 4,000
units per month, building up to a peak rate of over 13,000 units per month in
September. This plan, shown by the dashed line segments in Exhibit IV, would
result in substantially lower anticipation stocks. The average inventory would
be 3,700 units, with 3,400 units safety stock, or 300 units seasonal anticipation
stock. At $45 a unit, the cost of seasonal stock under this plan would be only
$13,000 per year, a saving in inventory cost of over $400,000 per year.
EXHIBIT V. MONTHLY ENDING INVENTORY
(In units)
January
5,830
July
18,830
February
r
10,170
August
15,170
March
15,500
September
10,500
April
19,830
October
6,830
May
22,170
November
4,170
June
21,500
December
3,500
Average monthly inventory
12,800
Average safety reserve
3,400
Average
seasonal anticipation stock 9,400
The saving, of course, is not all net saving, since it is gained at the cost of
adding and laying off the equivalent of some 9,200 units of production capacity.
If this were, say, a chemical plant operating well under capacity and the varia
tion from 4,000 to 13,200 units of production a month could be managed by
adding and then laying off some 100 semiskilled men, the saving in inventory
cost — equivalent to $4,000 per man hired and released — might well justify the
change. On the other hand, if the change in operating levels involved adding
and laying off some 1,000 to 1,500 employees of various skills, the inventory
saving might fall short of offsetting the hiring, training, and layoff costs, not
to speak of its effect on community relations. Under these circumstances, the
change might not be worthwhile.
This alternative production plan, of course, calls for substantially increased
plant capacity — nearly 60% more — for the same average throughput. If the
capacity were not available and had to be added, or if it would be gained at
the cost of overtime or secondshift premiums, or additional equipment in
stallations, the simple cost calculation just outlined would have to be extended
to include these extra costs and investments (not a difficult task if the proce
dures are well laid out).
By making similar trial calculations under other operating patterns, one can
quickly get a picture of the influence of operating pattern on cost, and can
arrive at a pattern which comes close to giving the minimum overall cost.
This plan then represents the basis for procurement, employment, and inven
tory control during the coming months until new forecasts call for an adjust
ment.
234
INVENTORY CONTROL
The operating plan summarized in Exhibit VI is essentially a minimumcost
plan, under the conditions that: (a) inventory costs are $45 per unit; (b) the
cost of hiring and training an employee is $300 (typical of many industries);
(c) a change of 750 units in the monthly rate of output requires employment
or release of 100 men. The cost of seasonal inventory equals 2,150 units
(average seasonal anticipation stocks) X $45 per unit, or about $97,000. The
plan calls for varying the production rate from a low of 5,000 units per month
to a maximum of 1 1,000 units — a change of 6,000 units; this requires hiring and
training 800 new employees at a cost of $240,000. (If the hiring and subsequent
layoff of 800 employees is considered an undesirable employment variation, the
solution must be sought within whatever are set as the feasible or tolerable
levels.)
Thus, under the plan in Exhibit VI the total of seasonal anticipation inventory
stocks and hiring and training costs is $337,000. This represents a net saving
of nearly $90,000 per year compared either with the uniform production plan
or the alternative plan in Exhibit IV. (With the hiring and training costs taken
into account according to the conditions assumed for Exhibit VI, the alterna
tive plan with its extreme employment variation comes out about the same
as the uniform plan.)
Advanced Techniques. Sometimes the problem of planning production
to meet seasonal demand is too complicated for simple graphical tech
niques, and more specialized techniques are needed. One of these is linear
EXHIBIT VI.
MINIMUM
OVERALL
COST PLAN
(In units)
Endofmonth
Monthly
inventory
Sales
production
(including
Month
forecast
plan
safety reserve)
January
6,000
5,500
3,000
February
4,000
5,000
4,000
March
3,000
5,000
6,000
April
4,000
5,000
7,000
May
6,000
5,200
6,200
June
9,000
11,000
8,200
July
11,000
11,000
8,200
August
12,000
11,000
7,200
September
13,000
11,000
5,200
October
12,000
11,000
4,200
November
11,000
10,800
4,000
December
9,000
8,500
 3,500
Average monthly inventory
5,550
Average safety reserve
3,400
Average seasonal anticipation stock
2,150
programming. The problem just described might have been attacked by
linear programming methods in order to cut through the repeated trials to
a good solution, but this approach was not necessary because trial and
GUIDES TO INVENTORY POLICY: III. ANTICIPATING FUTURE NEEDS 235
error did not involve a prohibitive amount of time and effort. Linear
programming has been found useful in circumstances where the problem is
complicated, for instance, by one or more of these conditions:
Several product lines using the same facilities or staff.
Possibilities of planned use of overtime to meet peak needs.
Need for considering extrashift premiums.
Several stages in manufacturing, with seasonal storage possibilities between.
A number of alternate plants, with different cost and employment situations,
to meet demand.
Joint planning of plant operations and of the assignment of branch ware
houses to the plant.
When the seasonal planning problem is attacked as a linear programming
problem, the objective is to minimize the total of costs incurred in carry
ing inventories forward in slack periods to meet future sales peaks, chang
ing the production level to meet sales requirements, or resorting to over
time. The objective has to be reached within the limitations imposed by:
(a) capacity restrictions on the amount which can be produced at normal
or overtime rates in any month; (b) the requirement that inventories in
each line or product be planned large enough to meet sales requirements;
and, possibly, (c) the amount of variation that can be tolerated in the
planned production rate.
Illustrations of production planning problems formulated in linear pro
graming terms can be found in technical literature on the subject.
CONTROLLING PRODUCTION
Once the production plan has been made, it and the sales forecast
dictate a sequence of planned inventory balances. However, as sales
experience accumulates, actual stocks will fall below or exceed the
planned balances. The minimum inventory balance or safety stock which
has been (or should have been) set up will absorb the immediate effects
of departures of actual sales from forecast, but it will be necessary to keep
adjusting production plans periodically to bring inventories into line. The
size of the needed safety stock, it should be emphasized, depends on the
way production adjustments are made.
The task is comparable to that of adjusting production in the face of
demand fluctuations, described in the preceding article in this series. There
it was pointed out that methods used generally take this form: adjusted
production = original production plan (or forecast sales level) ± some
fraction or part of the deficiency or excess of inventory compared with
"normal" or "par." The idea is to keep adjusting production to bring
236 INVENTORY CONTROL
inventory back into balance in the face of fluctuations in demand. If the
fraction is large (close to one), production is made very responsive to
sales fluctuations, and the inventory needed is smaller. If the fraction is
small, the inventory acts to absorb sales fluctuations, and must be larger;
production changes from original plan are smaller.
Production plans to meet seasonal sales have to be kept in adjustment
in much the same way, and basically similar control systems can be used.
In this case, the original plan is the production plan (e.g., Exhibit VI)
worked out to meet seasonal sales. The "normal" inventory is not a fixed
level, as in the other case, but varies from month to month; it is the
planned inventory of Exhibit VI, including the safety stock. The steps to
take in planning production are these four:
1. From a study of forecast errors or possible differences between sales
and forecast, and of costs of holding inventory and changing production,
choose the desired fraction or rate of adjustment in production and the cor
responding safety stock, using methods of the type described in the preceding
article.
The choice of safety stock does not involve production levels, just the costs
of changing production and holding stocks, along with anticipated forecast
errors.
2. Add the safety stock so chosen to the cumulative sales forecast month
by month to get the accumulative production required.
3. Plan production period by period to meet requirements, as described
earlier.
4. Periodically adjust the planned production by the specified fraction
(chosen in Step 1) of the departure of actual inventories from the plan for the
period.
For those interested in the actual working through of a problem, Ex
hibit VII shows the mathematical expression of the production control rule.
RELATIVE IMPORTANCE
The relative importance of anticipation stocks and of lot size stocks
and fluctuation stocks (described in the previous articles) will differ from
case to case. A study of sales and production characteristics is basic in
finding out what inventory functions are important, and what the sig
nificant costs and policies related to these functions happen to be.
GUIDES TO INVENTORY POLICY: III. ANTICIPATING FUTURE NEEDS 237
EXHIBIT VII. PRODUCTION CONTROL RULE EXPRESSED
MATHEMATICALLY
The rule can be written formally as :
Pi = Pt± k{lU  /«_i); < k ^ 1
where:
Pi = adjusted production plan for period i
Ft = original production plan for period i
/i_i = planned closing inventory for period i — 1
/i_i = actual closing inventory for period i — 1
k = fraction of inventory departure adjusted for in production
P* and it are chosen by the methods described earlier to minimize total inventory
and operating costs and meet the production requirements:
Ri = Fi J Si
where :
Ri = production required up through period i
Fi = accumulated forecast of sales through the period i
Si = safety stock needed for period i
The safety stock, Si for each period is in general proportional to expected forecast
errors and related to the value of k that is chosen. Thus, if forecast errors from
period to period are independent,
\ 2k
<ti being the standard deviation of forecast errors for the period; and A a parameter
which depends on the percentage of customer orders which management desires to
fill directly from stock (typical values range from 1.3 to 2.5, corresponding to 90%
and 99% protection).
SALES CHARACTERISTICS
Sales characteristics which strongly influence the production and in
ventory control system (and the relative importance of the different inven
tory functions) include:
1. The unit of sales — Are sales made in dozens, tons, or carloads? Planning
must be done in terms of this characteristic unit. It is obviously not enough,
for example, to have several tons on hand if the usual unit required is a carload.
2. The size and frequency of orders — Are there a few large orders each day
or week, or a steady stream of small orders? This is related to the question of
unit of sales, but the same total volume sold in a large number of small orders
can characteristically be supported by substantially less inventory than if sold
in a few large orders, unless special measures are taken to reduce the uncer
tainty about the time when individual large orders will be placed.
3. Uniformity or predictability of sales — Do sales show predictable seasonal
fluctuations? Or do they show large shortterm fluctuations, uncontrollable or
238
INVENTORY CONTROL
selfimposed (as by special sales campaigns)? Handling large, unpredictable
fluctuations requires flexibility and additional capacity in inventory production
as well as carefully designed rules for adjusting or controlling inventory balances.
But where fluctuations are predictable, advance planning techniques can be used.
4. Service requirements or allowable delay in filling orders — Where allow
able delays are small, inventories and production capacity must be correspond
ingly greater; care is required to be sure the control system is really responsive
to needs.
5. The distribution pattern — Do shipments go direct from factory to cus
tomer, through field warehouses, through jobbers, retailers, or consignment?
The more stages there are, characteristically, the more inventory is required.
Field inventories in fact serve basically to improve service to jobbers or re
tailers and thereby to remove from them some of the burden of keeping stocks.
Where the product moves through several stages of handling from factory
to ultimate consumer, prompt reports or estimates of movement, as close to
the consumer level as possible, are important in minimizing the amount of
uncontrollable fluctuation in demand which the factory has to contend with.
Often the reordering habits of retailers and jobbers can seriously exaggerate
the basic uncertainty in consumer demand for a product, and thereby com
pound the inventory and production control problems of the plant.
6. The accuracy, frequency, and detail of sales forecasts — Fluctuation stocks
exist basically because forecasts are not exact. Thus the inventory problems of
a business are directly related to its inability to forecast sales with precision.
This does not mean that lack of precise sales forecasts is an excuse for sloppy
control. Sometimes it is more economical to accept the forecasting uncertain
ties and stick to the plan, whether it means overproduction or underproduc
tion, than to pay the price in inventories or production fluctuations. But the
responsibility of forecast errors for inventory needs should be clearly recog
nized, and the control system should be adapted to the type of forecasts that
are possible.
PRODUCTION CHARACTERISTICS
The production characteristics which influence the scheme of produc
tion and inventory control are:
1. The form of production organization — Jobshop type organization is an
expensive way of getting flexibility; a company using it should be sure it really
needs that degree of flexibility. The inventory and production control scheme
can be considerably simpler under a productline organization than in a job shop.
2. The number of manufacturing stages — Where a number of stages in man
ufacturing exist, the inventory control scheme must be set up to take advantage
of differences in cost and obsolescence risk which are likely to exist.
3. The degree of specialization of the product at specific stages — Is each end
product distinct from the raw material stage on, or are the different products
more or less the same up to the final processing, assembly, and packaging?
Where the latter is true, economies are often possible in keeping the right
GUIDES TO INVENTORY POLICY: III. ANTICIPATING FUTURE NEEDS 239
balance of stocks in the semifinished state and by simplifying the control and
scheduling of preliminary stages where the types of product are not diverse.
4. Physically required processing times at each stage — Processing times affect
the length of delay, after issuing a replenishment order or adjusting a produc
tion rate, before the action becomes effective. The length of this delay, in turn,
directly influences the size of the inventory needed.
5. Capacity of production and warehousing stages — Capacity obviously af
fects the size and frequency of reorder.
6. Production flexibility — How rapidly can management vary production
rates, shift personnel among product lines or departments, and change equip
ment from one product to another? Management of inventories and production
control are basically a question of striking a balance among production flexi
bility and capacity, inventory levels, and customer service needs. No company
is free to pick all three at will. A realistic inventory control system must be
set up to recognize the limitations in flexibility which exist.
7. Kind of processing — Are batches of materials of a certain size needed in
production? If so, the quantities and combinations must obviously be taken
into account in scheduling for production.
8. Quality requirements, shelf life limits, or obsolescence risks — These set
important upper limits on the extent inventories can be used to buy flexibility
and free production operations from fluctuations in demand.
These sales and production characteristics cannot be readily distinguished
as having one type of effect or another on the production planning
and inventory control scheme. Nor is it true that one type of characteristic
dictates one approach while another kind of product always requires
something else. However, the job of setting up a sound production
and inventory control system is not just a job of setting up the right
clerical routines and staff organization; it is a research job to find out
how the product sales and production characteristics can be exploited to
get an economical balance between production flexibility, inventory in
vestment, and customer service.
ROLE OF FORECASTING
The need for estimates of future sales to control inventories is clearest
in the case of anticipation stocks, but it exists in the case of the other
functions as well. Whether forecasts are needed or possible is not the
question; they are made formally or informally every time a decision is
made whether to build or replenish an inventory. The question is whether
the necessary forecasts are being made as well as they might be if formally
recognized and if available statistical and market research techniques were
used. Without going into the methods of forecasting, which is a consider
able subject of its own, the following points are significant here :
Economical inventory plans depend on realistic estimates of need — not just
240 INVENTORY CONTROL
sales goals or quotas. Even so, there are bound to be forecasting errors — and
the bigger the possible errors, the bigger the inventories must be to guard
against them.
A single forecast figure, without specifying the estimated error or limits of
error, is not enough. Sometimes the need may be met by a maximum sale
forecast indicating the upper limit of demand which the production or dis
tribution organization will be required to service.
To estimate the limits of error requires a comparison of past forecasts and
sales — and often this is hard to do, either because the earlier forecasts were
made informally, or the records were discarded and hopefully forgotten.
In any event, forecasting errors bear so importantly on inventory economy
that to keep the control system up to date requires systematic review of past
errors and effort to improve the forecasting method.
PRODUCTION SCHEDULING
The task of translating inventory policy into practice, of reacting to
demand as it materializes and utilizing the inventory balances and planned
production capacity, is a function of production scheduling. Considerable
effort has gone into the development of techniques — board displays, filing
systems, card systems, and so on — to facilitate scheduling and control of
progress on orders scheduled. These techniques can be extremely useful
if they are adapted to the nature of the product and manufacturing facili
ties and if they are used in a framework of selfconsistent inventory
balances and production operating levels. The essence of the control
problem is setting this framework in the light of management policy, not
making the actual schedules.
Conventional scheduling methods are often worked out to cope with
the complexities of jobshop production, where each order is unique and
no set sequence of operations exists.
Scheduling operations in this way through a large number of stages or
departments is difficult. Fortunately, however, the need for so doing is
not nearly so common as one might gather. Many businessmen, in dis
cussing inventory and production control, give the impression that their
organization with its large product line — whether several hundred or
several tens of thousands — is saddled with jobshop operations from top
to bottom. They look with longing toward the lower operating costs and
simpler management problems of assemblyline operations. They frequently
fail to recognize that almost all products and product lines are capable
of being manufactured under a wide range of organizational forms inter
mediate to the extremes of either pure jobshop or assemblyline operation.
Taking advantage of this latitude has been a source of considerable
operating economy in some businesses — and could be in many more.
GUIDES TO INVENTORY POLICY: III. ANTICIPATING FUTURE NEEDS
241
CONTROL SYSTEMS
A comprehensive inventory control system should be closely coordi
nated with other planning and control activities, such as sales forecasting,
cash planning, and capital budgeting, since it affects all of these activies
in many ways. The specific steps and timing will vary from one company
to another, depending on product and process requirements, but the es
sentials of an inventory control system can be grouped into three broad
classes: longrange inventory planning, shortrange planning, and schedul
ing.
EXHIBIT VIII. SCHEMATIC DIAGRAM OF INVENTORY PLANNING
LONGRANGE
PLANNING
ANALYSIS
POLICY
DECISIONS
REQUIRED INVESTMENT
RETURN,
SERVICE POLICY,
EMPLOYMENT POLICY,
ETC.
SHORTRANGE
PLANNING
ANALYSIS
SHORTRANGE
OPERATING
DECISIONS
LONGRANGE
OPERATING
DECISIONS
CAPITAL FACILITIES,
SALES PROGRAMS,
ETC.
GENERAL PRODUCTION
PLAN,
EMPLOYMENT LEVEL,
INVENTORY BALANCES,
ETC.
PRODUCTION
SCHEDULING
Exhibit VIII shows the three basic planning functions in boxes, with the
arrows indicating the flow of information to and from the analysis. More
specifically:
1. The longrange plan makes use of: (a) sales forecasts, with error or
range estimates, and (b) preliminary policy decisions on capital allocation and
value and on the amount of risk to be assumed. The purpose is to show the
implications of policy choices so they can be refined and sharpened, and then
to provide a basis for longrange operating decisions concerning construction,
purchase, and sale of facilities, adjustment of sales and promotion programs,
and so on. The analysis results may also lead to further forecasting effort by
showing the production and capital costs resulting from poor forecasts.
2. At the intermediate stage, the shortrange plan uses as its "raw materials"
or inputs: (a) the results of policy decisions, (b) shortterm demand fore
casts, (c) existing facilities and manpower, and (d) inventories. The outputs
are bases for shortrange operating decisions — the general production plan to
follow, adjustments in the employment rate, corrections in inventory balances.
3. Finally, within this framework scheduling can react to demand as it
actually materializes.
Chapter 1 2
RELATED LOT SIZE AND
DYNAMIC PROBLEMS
XI v.
An Investigation of Some Quantitative
Relationships Between BreakEven Point
Analysis and Economic LotSize Theory*
Wayland P. Smith
Two common tools utilized to evaluate the economic potential of alter
native ways of performing a specified task are (A) BreakEven Point
Analysis, and (B) Economic LotSize Theory.
These two techniques have become the basic devices around which
many courses in Engineering Economics have been built over the past 40
years. Most engineering undergraduate students at some time are exposed
to these two fundamental methods. Industrial Engineers, in particular,
find a persistent recurrence of problems that utilize one or the other of
these models in their solution.
In retrospect one wonders why these two schemes have never been com
bined into one common model. On the other hand, they have primarily
* From The Journal of Industrial Engineering, Vol. 9, No. 1 (1958), 5257. Re
printed by permission of The Journal of Industrial Engineering.
242
AN INVESTIGATION OF SOME QUANTITATIVE RELATIONSHIPS 243
been developed to solve two problems that at least on the surface appear
to be quite divergent. The purpose of this paper is to show that these
problems are not divergent, that they have much in common, and that
they are actually very much interrelated. In short, a single mathematical
formulation will be developed that relates the two theories.
Before presenting such a combined formulation, a brief review is pre
sented of the two basic techniques in very simple form. The purpose of
this review is twofold — 1. to reestablish the assumptions on which these
two techniques are based, and 2. to establish a common set of symbols
that will be helpful for the combined formulation.
A SIMPLIFIED BREAKEVEN POINT MODEL
The historical development of breakeven analysis and the breakeven
chart would be exceedingly difficult to trace. Like many other techniques,
it was simultaneously created and developed by many people with the
rapid growth of the scientific management movement during the early years
of the twentieth century. It was one of the techniques that was quickly
exploited.
Certainly Dr. Walter Rautenstrauch was one of the principal developers
and proponents of this technique (7). It is to be found in practically all
books which deal with engineering economy problems (1) (2) (3) (4)
(10) (13).
Generally, it is treated in its most simple form for solving shortrange
problems and the timevalueofmoney aspect of the problem is omitted.
This is not always the case, and more sophisticated methods involving
different basic assumptions have been developed. One of these is the
MAPI formulation (6) (9).
Although many types of problems lend themselves to breakeven analysis,
a specific type of problem is of concern here — the economic comparison
of two alternative methods of performing a task (actually any number of
alternatives can be compared simultaneously by this procedure).
Costs for each alternative are divided into two categories and called
either fixed costs or variable costs. The fixed costs are those costs that
occur only once during the life of the alternative. Once these costs have
been expended, they are not recoverable. Examples of such costs are:
1. Set up and tear down cost.
2. Cost of special tooling.
3. Paper work and clerical cost required to schedule a job.
In other words, the fixed costs are those costs that are not incurred by
every unit of production. If we were to plot cost versus the number of
units produced, these costs would appear as step functions. In the most
244 INVENTORY CONTROL
simple problem, only those fixed costs that occur prior to the first unit
of production are included. This is frequently done when the other fixed
costs are assumed to be negligible.
Conversely, variable costs are those costs that are incurred by every unit
of production. Examples of such costs are:
1. Direct labor cost per piece.
2. Direct material cost per piece.
3. Depreciation.
Once both types of costs have been identified for each alternative, it is
possible to write a total cost equation for each one of the alternatives in
terms of production quantity.
Total cost = (summation of all fixed costs) + (the production quantity)
X (the summation of all variable costs)
Where: d = total cost of the /th alternative
f xi = an initial cost of the /th alternative
Fi = summation of all initial fixed cost for the /th alternative
v xi = a variable cost of the /th alternative
Vi = summation of all variable cost for the /th alternative
N = the quantity to be produced Eq. 1 .
Then d = (hi + f 2i + '••)+ N (v 14 + v 2i + • • • )
Or Ci=Fi + NVi
If Ci is plotted against N for each alternative, a simple graphical relation
ship results (Fig. 1). Fi becomes the y intercept and V { becomes the
slope of the straight line that this equation represents. If the two alterna
tives have been selected such that the one having the lower F 4 has the
higher Vi, then the two lines will intersect at some positive value of N.
This point is called the breakeven point. It represents the quantity at which
the total costs are the same for both alternatives. When the demand for
the product is greater than this breakeven quantity, then the alternative
with the lower Vi is more economical. When the required quantity is less
than the breakeven point, then the alternative with the lower F 4 is the
more economical.
In a quantitative manner this analysis tells us what we recognize intui
tively — that a production method with relatively low fixed costs and high
variable costs will be more economical at lower quantities while a produc
tion with relatively high fixed costs and low variable costs will be more
economical at higher quantities.
It also reminds us that no single production method is best for all
quantities of production. Since minor and major methods variation are easy
to conceive through changes in machines, tools, men, motion patterns, etc.,
AN INVESTIGATION OF SOME QUANTITATIVE RELATIONSHIPS
245
8
1 F 2
decision rules —
if N<Nb, select 1
if N>N b/ select 2
N ^
total quantity required, pieces
Figure 1 . A BreakEven Chart for Two Alternatives
it should not be too difficult to imagine an almost infinite number of pos
sible production methods from which to choose — each with its own small
range of production quantities over which it is more economical than any
other alternative (Fig. 2).
a simplified production economic
lotsize model
Like breakeven analysis, economic lotsize theory was developed during
the period from 1910 through 1930. Its historical development has been
clearly traced by Fairfield E. Raymond in his monumental treatise on the
subject ( 8 ) . Since this book tells the story so well and contains a complete
bibliography, it would be superfluous to repeat it here. However, a table
from this text showing the development of economic lotsize formulas has
been reproduced here because it is so revealing.
246
INVENTORY CONTROL
TABLE 1. EQUATIONS EMPLOYED TO SHOW THE STAGES
IN THE DEVELOPMENT OF FORMULAE FOR
ECONOMIC LOT SIZES
First
appear
ance Form
Authorities
Approx.
date of
record
1912
1915
1917
1917
1918
1918
1923
1924
Cubic equation not publ
shed
urance,
G. D. Babcock
F. W. Harris
D. B. Carter
General Electric Co.
J. A. Bennie
P. E. Holden
K. W. Stillman
Benning and Littlefield
J. M. Christman
G. H. Mellen
Eli Lilly & Co.
S. A. Morse
W. E. Camp
Heltzer Cabot Co.
N. R. Richardson
H. T. Stock
B. Cooper
G. Pennington
E. T. Phillips
W. L. Jones
J. W. Hallock
•k
E. W. Taft
G. Pennington
F. H. Thompson
R. C. Davis
C. N. Neklutin
A. C. Brungardt
P. N. Lehoczky
1912
/ PS
Special adaptation
1915
1922
1922
1923
1924
1925
1925
1917
/ PS
\ Cl
1917
1922
1924
1927
c / P ' s ' k
\ cV + fi)
where
fi = allowances for ins
storage costs, etc.
1923
1926
/ PSD'
\ ci(D'S)
1927
1927
1929
1929
p \p2
Q= +J+
c \ c 2 c
PSD'
i(D'S)
■k'b
1918
1 PSk
h
1927
1923
1926
1929
c ! psk
1923
\ ci + (ra + c) •
t p 'Si
1927
The operations research movement stemming from the success of opera
tions evaluations groups in the various services during World War II has
given a rebirth to economic lotsize theory. It is interesting to note, how
ever, that the recent writings of many operations researchers seem oblivious
AN INVESTIGATION OF SOME QUANTITATIVE RELATIONSHIPS
247
of the work done during the 1920's. At least there appears to be some
hesitancy towards acknowledging this earlier work (5) (11). Economic
lot size as it relates to inventory control has also been treated recently
by T. M. Whitten in the theory of inventory management (12).
decision rules
if N <N<00, select 6
if N, ,<N<N, _, select 5
b4 bo
N
bl
N
b2
N,
N,
N.
b5
b3 b4
total quantity required, pieces
Figure 2. BreakEven Chart for Multiply Alternative Case
In the production of goods when the rate of production exceeds the
rate of demand, the production must proceed in batches if the maximum
production rate of the process is to be fully utilized. Otherwise, a ponderous
growing stock pile will soon exist. The question arises as to the size of
these batches or length of the run or the number of batches or lots to be
started each year (in other words, four ways of stating the same problem).
In order to achieve the lot size that is most economical, two opposing
cost elements must be considered. On the one hand, there are the costs
that are incurred for each new lot, namely, setup cost of the process
including necessary scheduling, handling, paper work, etc. These costs
are relatively proportional to the number of lots and would tend to make
the lot size large. On the other hand, there are the costs incurred due to
the storage of items necessitated by the production rate being higher than
248
INVENTORY CONTROL
the demand rate. When long production runs are made, the amount of
storage space required and the amount of money involved in inventory
would increase. These costs would tend to make the lot size small.
Economic lotsize theory as does breakeven analysis involves the
development of the total cost equation for any specific situation of this
kind.
Total cost = (the number of lots) X (the preparation and setup costs)
+ (the production quantity) X (the sum of all variable
costs) f (average inventory quantity) X (the inventory cost
per unit during a given time period) X (number of time
periods during the demand period) Eq.2.*
Where d = total cost of the ith alternative
N = total quantity to be produced
n = lot size quantity
Fi = sum of preparation and setup costs for the ith alternative
Vi = sum of the variable costs for the /th alternative
Pi — production rate (after setup) costs for the /th alternative
D = the demand rate
~(\ N/P i) = the average inventory quantity (assuming a constant demand
2 rate)
/ = sum of storage and inventory costs per unit during a given
time period
Then C«
F t + NV<
=(")
or c '= JV [®(l^) nl+ ^ 0+FiB " 1
If Ci is plotted against n for a specific alternative, a simple graphical
relationship results for any specified value of D. This represents the graph
ical sum of the three costs in Eq. 2. (Refer to Fig. 3.)
By setting the first derivative of cost with respect to the lot size equal
to zero, it is possible to find the lot size which yields minimum cost.
dC t
dn
= o^ N \L(l.L)h
[2\D Pj n 2
Eq. 3.
* This only is true where P > D.
AN INVESTIGATION OF SOME QUANTITATIVE RELATIONSHIPS
249
n ==
2F t
fek)
Eq. 4.
If this value of n is substituted back into Eq. 2., a total cost equation
results which is the minimum cost for any value of N.
t*"GXK).
2Fi
Gk)
Eq. 5,
min^N l(— V« +
iVK,
When this result is compared with Eq. 1., there is considerable similarity.
The variable cost term is identical. The coefficient of the fixed cost term
in the economic lotsize formulation is considerably more complex than
that of the breakeven formulation. In the breakeven formulation, this
coefficient is unity.
It should be apparent from the foregoing simple formulation of economic
lot size that the concept of alternative manufacturing methods is totally
ignored. Economic lotsize theory as developed here assumes one and only
one manufacturing method. While this may be realistic enough after pro
duction methods have been established, it is not realistic at the time the
manufacturing method is being established.
Thus two methods have been explored. Both of these methods purport
to select the best method of doing a job. But these two techniques are based
upon assumptions that are radically different. Breakeven analysis ignores
inventory cost and assumes a single set up while economic lotsize theory
ignores the possibility of different manufacturing methods with their
accompanying differences and fixed variable cost patterns.
250
INVENTORY CONTROL
THE COMBINED MODEL
There are two possible ways of approaching combined formulation of
breakeven analysis and economic lotsize techniques. The simpler case
is the discrete case which assumes a limited finite number of possible
production alternatives. The more complex case is the continuous one which
assumes an infinite number of alternative production methods. The first
case selects the optimum method from several stated possibilities. The sec
ond case selects the cost pattern of the optimum alternative and then
searches for the method which actually matches this cost pattern. Only the
first case is discussed in this report.
To fully explore the discrete case, we will examine a problem involving
three alternatives: namely, the ageold problem of whether it is better to
use an engine lathe, a turret lathe, or an automatic lathe to perform a
specified manufacturing operation.
decision rule 
select n such that
dC/dn =
lot sizes, pieces per lot
Figure 3. Graphic Formulation of Economic Lot for
A Single Alternative
AN INVESTIGATION OF SOME QUANTITATIVE RELATIONSHIPS 251
To make this combined formulation as realistic as possible, the follow
ing values have been assumed:
1. Sum of costs that are incurred each time the job is set up (FJ.
a. Engine lathe, F 1 = $1
b. Turret lathe, F 2 = $30
c. Automatic lathe, F 3 = $70
2. Sum of all variable costs including direct labor, direct material, and man
facturing expense (T^).
a. Engine lathe, V 1 — $.20 per unit
b. Turret lathe, V 2 = $.10 per unit
c. Automatic lathe, V 3 = $.05 per unit
3. Sum of all storage and inventory costs per unit of production when stored
for one month. This amount is the same for all three alternatives (/).
/ = $.25 per month per unit.
4. Production rate (Pi).
a. Engine lathe, P 1 = 5,000 pieces per month
b. Turret lathe, P 2 — 15,000 pieces per month
c. Automatic lathe, P 3 — 45,000 pieces per month
All of these values are inherent in the particular production method and storage
system. In a sense, these are fixed values. In addition, we must know or as
sume values relative to customer demand. We wish to examine what happens
at different levels of customer demand.
To make this formulation more realistic the following levels have been
assumed:
5. Demand rate (D).
Case No. 1, D 1 = 100 pieces per month
Case No. 2, D 2 — 1 ,000 pieces per month
Case No. 3, D 8 = 5,000 pieces per month
Now it is possible to calculate the total cost equation for each alternative
according to Eq. 5. This is done in the three steps shown as follows:
Step No. 1 (the basic equation)
Engine lathe
mm
in = N . (—  — — ^ 1 + 2N Eq. 6.
\.4\Z>, 5,000/
Turret lathe
min = N ^ (— — — ^ 30 + AN Eq. 7.
C2 \AA 15,000/
252 INVENTORY CONTROL
Automatic lathe
mm
C3
N . (—  — ^ 70 + .057V Eq. 8.
V 4VD, 45,000/ H
**p»o.2a*K» =y ii(± r ±y
and find the K value for each alternative at each demand rate
— 9 cases in all.)
Z>i
£>2
£>3
£i .049
.014
K 2 .274
.084
.031
#3 .42
.132
.056
(substitute K values into
Eq.
6., 7. and 8.
and combir
D 1
D 2
P 3
min .249N
Ci
.2147V
.2N
min .3747V
c 2
4847V
4311/
min .4257V
c 3
.1827V
406N
It is now possible to plot the total minimum cost of each alternative
against the demand rate. Figure 4 shows this relationship.
When a smooth curve is drawn through the points for each alternative,
three breakeven points become apparent. The engine lathe is the most
economical method when the demand rate is greater than and less than
450 pieces per month. The turret lathe is the most economical when the
demand rate is greater than 450 pieces per month and less than 900 pieces
per month. The automatic lathe is the most economical when the demand
rate is greater than 900 pieces per month.
How do these answers compare with those obtained from the typical
breakeven analysis? In the first place, it should be pointed out that these
answers are entirely independent of the total quantity required (N) . Thus
the answer obtained is much more general. To compare these answers with
the standard breakeven analysis, it is necessary to assume various values
of N.
AN INVESTIGATION OF SOME QUANTITATIVE RELATIONSHIPS
253
Substituting known values into equation 1 yields the following total
cost equation for each alternative according to the breakeven analysis
model which has been developed previously:
Engine lathe,
Turret lathe,
Automatic lathe,
d = 1 + .IN
C 2 = 30+ AN
C 3 = 70 + .05N
Eq. 9.
Eq. 10.
Eq. 11.
8
.25N r
20N
.15N 
ION 
05N
C (automatic lathe)
decision rules
if 0<D<450 / select 1
if 450<D<900 / select 2
if 900<D<a>, select 3
lit
:•;
1000 2000 3000
demand rate, pieces per month
4000
5000
Figure 4. Combined Formulation for Three Alternatives
When the C t versus N curves are plotted, three breakeven points are once
again apparent. The engine lathe is the most economical method when the
quantity required is greater than and less than 290 pieces. The turret
lathe is the most economical method when the quantity required is greater
than 290 pieces and less than 800 pieces. The automatic lathe is the most
economical when the quantity required is in excess of 800 pieces. This
relationship is shown in Figure 5.
254
INVENTORY CONTROL
120 I
100
80
■S 60
o
40
20
decision rules —
if 0<N<290, select 1
if 290<N<800, select 2
if 800<N<00, select 3
1000
total quantity required, pieces
Figure 5. BreakEven Chart for Three Alternatives
COMPARISON OF RESULTS
The next problem is to find a convenient and graphic way of portray
ing the errors that would have been generated in this specific problem
if the simple breakeven method had been used to solve the problem.
Or to state it another way, what savings would have been derived by using
the more complicated, combined model in determining the best pro
duction method?
A glance at the decision rules which are established in this example by
the breakeven model and the combined model is revealing because in most
situations the two methods result in different courses of action rather
AN INVESTIGATION OF SOME QUANTITATIVE RELATIONSHIPS
255
than the same course of action. Figure 6 clearly reveals the disparity be
tween the two methods. Listed below are the types of errors that can be
made. These are shown graphically in Figure 6. In each case the area over
which the error can be made is a rough approximation of the probability
of this type of error. The actual loss at a specified point in the Figure 6
matrix may be determined from Figure 4.
TYPES OF ERROR
Type of Error
Breakeven
Model Chooses
Combined
Model Chooses
1
Engine Lathe
Turret Lathe
2
Engine Lathe
Automatic Lathe
3
Turret Lathe
Engine Lathe
4
Turret Lathe
Automatic Lathe
5
Automatic Lathe
Engine Lathe
6
Automatic Lathe
Turret Lathe
only the shaded areas yield the same
decision by both formulations.
2000
1600
1200
12 3 4
demand period (N/D), months
Figure 6. Error Space Diagram
256 INVENTORY CONTROL
Generally a sales forecast would be able to limit to some degree the
space in this figure that is actually pertinent. Let us say in this example
that we are reasonably certain that the total volume required will be be
tween 1,200 and 2,100 pieces and that the demand rate will be between 200
and 700 pieces per month. If this were the case checking Figure 6, it is
apparent that the breakeven model would give us the wrong decision over
this entire space. According to the breakeven model, we should always use
l] the automatic lathe. According to the combined model, we should either
use the engine lathe or the turret lathe depending on the point in this
space that actually occurs.
CONCLUSIONS
1 . The use of the simple breakeven model to decide between several alterna
tive methods of performing a task can give rise to serious errors.
2. These errors are particularly pronounced when —
a. The production rates are considerably higher than the demand rates.
b. The storage and inventory costs are significantly large.
c. There is a substantial difference between the ratio of setup cost and
storage cost for the several alternative methods.
3. The combined model is not significantly more difficult to work with when
modern computing techniques and equipment are considered.
References
(1) Bullinger, C. E., Engineering Economic Analysis, McGrawHill, New
York, 1950.
(2) Eidmann, F. L., Economic Control of Engineering and Manufacturing,
McGrawHill, New York, 1931.
(3) Grant, E. L., Principles of Engineering Economy, Ronald Press, New
York, 1950.
(4) Knoeppel, C. E., Profit Engineering, McGrawHill, New York, 1933.
(5) Lathrop, John B., "Production Problems Bow to Operations Research,"
SAE Journal, May 1954, p. 46.
(6) Orensteen, R. B., "Topics on the MAPI Formula," The Journal of In
dustrial Engineering, Nov.Dec. 1956, Vol. 7, No. 6, p. 283.
AN INVESTIGATION OF SOME QUANTITATIVE RELATIONSHIPS 257
(7) Rautenstrauch, Walter, and Villers, Raymond, The Economics of
Industrial Management, Funk & Wagnalls Co., New York, 1949.
(8) Raymond, Fairfield E., Quantity and Economy in Manufacture, Mc
GrawHill, New York, 1931.
(9) Terborgh, George, Dynamic Equipment Policy, McGrawHill, New
York, 1949.
(10) Thuesen, H. G., Engineering Economy, PrenticeHall, Inc., New York,
1950.
(11) Varnum, Edward C, "The Economic Lot Size," The Tool Engineer,
Nov. 1956, p. 85.
(12) Whitest, Thomson M., The Theory of Inventory Management, Princeton
University Press, Princeton, N.J., 1953.
(13) Woods, B. M., and De Garmo, E. P., Introduction to Engineering
Economy, Macmillian, New York, 1953 (2nd Edition).
Chapter 13
SIMULATION IN INVENTORY
CONTROL
xv.
99
Determining the ^Best Possible
Inventory Levels*
Kalman Joseph Cohen
INTRODUCTION
We shall be concerned here with the problem of how inventory levels
should be established. This is an important problem, for inventories play
a role in all phases of business and industry, whether retail, wholesale, or
manufacturing. Rather than considering all aspects of inventory problems,
however, we shall concentrate on the function of inventories in the dis
tribution of commodities, in particular, in the wholesaling operations, and
we shall not be concerned with the part that inventories play in manu
facturing.
In a distributional business, the reason for having any inventory is to
sell goods. Usually, you cannot sell anything to a customer unless you have
it in stock. These considerations would tend to make wholesalers or re
tailers carry large inventories, for the more items there are in stock, the
greater are the chances of having what customers want when they want it.
* From Industrial Quality Control (1958), 410. Reprinted by permission of In
dustrial Quality Control.
258
DETERMINING THE "BEST POSSIBLE" INVENTORY LEVELS 259
However, there are limits to the size of the inventories which any busi
ness would want to have, these limits arising because it costs money to hold
inventory. Inventories can be considered to be too large when the costs
of carrying the commodities exceed the benefits obtained from having them
in stock. On the other hand, inventories are insufficient when the additional
gains from having more inventory are greater than the additional costs which
would be generated. Somewhere in between there exists the best possible
level of inventory, i.e., that level of inventory which results in the largest
possible profit for the business. How to find the level of inventory which
is best is the subject of this article.
THE WAREHOUSE NETWORK !i
The particular work which we shall discuss was done for the Replace
ment Division of Thompson Products. This Division does no manufacturing,
so we need not consider the functions of inventories in manufacturing
operations. Replacement Division is a large distributor of automotive engine
and chassis parts for replacement use. This Division buys these parts in
bulk from various manufacturers and sells them to wholesalers or jobbers,
who in turn sell them to independent garagemen, car dealers, and fleet
operators.
Replacement Division has one central warehouse in Cleveland and 36
branch warehouses scattered throughout the United States. Most of the
sales are made at the branches. The central warehouse serves mainly as a
supply depot, from which all branch stocks are sent.
The inventory problem at the branch warehouses consists in determining
the best possible amounts of inventory which should be carried in stock.
The magnitude of this problem can be appreciated from the fact that
approximately 15,000 items are sold at each of the 36 branches, so there
are more than half a million branch warehouse inventory levels to be
established.
SALES DEMAND AT THE BRANCHES
If it were possible to say exactly how many parts would be sold each
week at every branch and how long it would take to get these parts to
the branches from the central warehouse, then it would be relatively easy
to specify the best possible inventory levels at each branch warehouse.
The problem that we have to deal with is considerably more difficult,
however, because of two unavoidable elements of uncertainty: uncertainty
about the sales demand at the branches and uncertainty about the time
required to send parts from the central to the branch warehouses.
260
INVENTORY CONTROL
The uncertainty in the sales demand for an item which occurs at the
branches is illustrated in Fig. 1, where we see a typical pattern in the
fluctuations of weekly demand. It looks as though we could say with some
degree of assurance that the sales demand for that part at the branch will
average ten units per week, but what it will be in any particular week, we
cannot say. Some orderliness can be derived from this chaos, however, if
we ask, not what the demand will be in any particular week, but how
frequently we can expect weekly demands of different amounts.
REPLENISHMENT
CYCLE TIME (WEEKS)
6
SUCCESSIVE BRANCH RECEIPTS
Figure 1. Variations of Demand for an Item at a Branch
If we look at the weekly demands for a part at a branch over a large
enough number of weeks, there will become evident a stable pattern of the
relative frequencies with which different levels of demand occur. Over a
long enough period of time, the pattern of weekly demand that we saw in
Fig. 1 will build up into the frequency distribution of weekly demand
shown in Fig. 2. Here we see that the most frequently occurring weekly
demand is the average demand of ten units per week, the next most
frequently occurring demand is nine units per week, and so forth. Further
more, a weekly demand of ten units occurs three times as frequently as a
weekly demand of five units, etc.
Statisticians apply the name "Poisson distribution" to describe the type
of frequency distribution shown in Fig. 2. For our purposes, we need know
only that the shape of the Poisson distribution is completely determined
by its average value, and that we can use this distribution to describe the
DETERMINING THE BEST POSSIBLE INVENTORY LEVELS
261
uncertainties in the weekly demand for a part at a branch warehouse.
Because of the fluctuations in sales demand, it sometimes happens that
a branch is out of stock on an item when a customer wants it. This results
in a lost sale, for when a customer cannot immediately obtain a particular
part from the local branch warehouse, he will usually purchase it from a
competitor. Thus, lost sales are lost business, and if a branch is out of stock
when an item is requested, Replacement Division foregoes the profit that
could have been made by selling that item.
THE REPLACEMENT SYSTEM
The branch warehouses are not autonomous in their operations. The
central warehouse maintains rigid inventory control over the branches, de
termining what their inventory levels should be and actually shipping all
stocks for the branches from Cleveland. Whenever a part is sold at a
branch, a copy of the sales slip is immediately sent to Cleveland, so that
the central warehouse can prepare an invoice for the customer. Cleveland
accumulates the sales slips from a branch for a whole week, and it then
prepares a replenishment shipment to restore the branch warehouse's in
ventory, replacing each part which has been sold during the week on a
oneforone basis. Thus, sales of all parts for a week are automatically re
placed on a weekly basis, and a replenishment shipment is sent each week
to every branch warehouse.
REPLENISHMENT CYCLE TIME
The period between the first sales slip for a week being sent from a
branch to Cleveland and the receipt of the corresponding replenishment
shipment at the branch can be called the replenishment cycle. Another
FREQUENCY
nd
16
18
20
2 4 6 8 10 12 14
WEEKLY DEMAND
Figure 2. Frequency Distribution of Weekly Demand for an
Item at a Branch
262
INVENTORY CONTROL
major element of uncertainty in the system is the length of the replenish
ment cycle, for this varies from week to week. Fig 3 shows a typical pattern
in the fluctuations of replenishment cycle times. While the average replenish
ment cycle time shown here seems to be three weeks, there are substantial
variations from one week to another. We can deal with the uncertainty of
2th WEEK
Figure 3. Variation of Replenishment Cycle Time for a Branch
FREQUENCY
4
1 ■
2 3 4
REPLENISHMENT CYCLE TIME
6 WEEKS
Figure 4. Discrete Frequency Distribution of Replenishment
Cycle Time
DETERMINING THE "BEST POSSIBLE" INVENTORY LEVELS 263
the replenishment cycle time as we dealt with the uncertainty in weekly
sales demand. That is, we shall ask not what the replenishment cycle time
will be in any particular week, but how frequently we can expect replenish
ment cycles of different lengths.
If we look at the replenishment cycle time over a large enough number
of weeks, there will become evident a stable pattern of the relative fre
quencies with which different lengths of replenishment cycles occur. Over
a sufficiently long period of time, the pattern of replenishment cycle times
that we saw in Fig. 3 will build up into the frequency distribution of
replenishment cycle times shown in Fig. 4. Here, we see that the most
frequent replenishment cycle time is three weeks, which occurs about 38
percent of the time. Replenishment cycles of two or four weeks each occur
about 24 percent of the time, and replenishment cycles of one or five weeks
each occur about 7 percent of the time.
Statisticians apply the name "normal distribution" to the type of fre
quency distribution shown in Fig. 4. For our purposes, we need know only
that the shape of the normal distribution is completely determined by its
average value and its standard deviation, and that we can use this distri
bution to describe the uncertainties in the replenishment cycle time at a
branch warehouse.
The uncertainties in replenishment cycles are actually somewhat more
complicated than those indicated by the distribution shown in Fig. 4. How
long the replenishment cycle will be in one week seems to depend on how
long it was in the previous week, since there is a positive correlation in
the lengths of successive replenishment cycles. Although over a long period
of time, the relative frequencies with which different replenishment cycles
occur is adequately described by the normal distribution, the positive cor
relation between successive replenishment cycles times means that the
magnitude of the weekly fluctuations will be lessened. The "correlated"
time series is much more sluggish than the "independent" time series. With
out going into complicated mathematical details, we can say only that the
positive correlation between successive replenishment cycles can be super
imposed on the distribution of replenishment cycle times, the result being
that statisticians would describe the uncertainties in replenishment cycle
time as a serially correlated, normally distributed random variable. (Since
we have momentarily lapsed into statistical jargon, we can mention in
passing that statisticians would similarly describe the uncertainties in weekly
sales demand as a Poisson distributed random variable.)
264 INVENTORY CONTROL
THE RELATION BETWEEN LOST SALES
AND INVENTORY LEVELS
Let us once again look at the schematic relation between the central and
branch warehouses. This is illustrated in Fig. 5, where we have introduced
roulette wheels to indicate the uncertainties in weekly sales demand and
replenishment of cycle time. These two elements of uncertainty make it
impossible for a branch always to have parts in stock when customers want
them. The frequency with which a branch is out of stock on items that
customers want depends upon the size of that branch's inventories, for the
branch warehouse inventories function as buffers which mediate the un
certainties of sales demand and replenishment cycle time. Therefore, a
necessary step in determining the best possible levels for branch inventories
is first to determine the relation between lost sales of an item and the level
of inventory for that part at a branch. 1
If sufficient historical data were available, it would be possible to deter
mine directly how lost sales depend upon inventory levels. The required
data were not available, however, so it was necessary to adopt an alterna
tive approach. Essentially, what we did was to reconstruct or simulate the
history of branch warehouse operations on a highspeed electric computer. 2
From this reconstructed history, we were able to obtain the required data,
and thus, the dependence of lost sales on inventory levels.
Technically, this simulation of history on a computer is called a "Monte
Carlo" approach, after the gambling casino of the same name. Anybody
who has watched the roulette wheels at Monte Carlo or Las Vegas can
understand the reasoning behind this approach.
1 An analytical method for determining the relation between average lost sales and
inventory level for the special case in which successive replenishment cycle times are
uncorrelated has been developed in William Karush, "A Queuing Model for an
Inventory Problem," Operations Research, Vol. 5, No. 5 (Oct. 1957), p. 693703,
and in Philip M. Morse, Queues, Inventories and Maintenance, New York: John
Wiley & Sons, Inc., 1958, p. 139146.
2 This Monte Carlo approach has been discussed in: (a) Kenneth C. Lucas and
Leland A Moody, "Electronic Computer Simulation of Inventory Control," p. 107
121 in Electronics in Action: The Current Practicality of EDP, Special Report No. 22,
American Management Association, New York, 1957. Figures 2, 4, 5, 8, and 9 were
taken from the Lucas and Moody paper; (b) Jack K. Weinstock, "An Inventory
Control Solution by Simulation," p. 6571 in Report of System Simulation Sym
posium (Sponsored by the American Institute of Industrial Engineers, The Institute of
Management Sciences, and the Operations Research Society of America, held in con
junction with the 8th National Convention of the American Institute of Industrial
Engineers, New York, May 16, 17, 1957), 1958; (c) Andrew Vazsonyi, "Electronic
Simulation of Business Operations (The Monte Carlo Method)," Second Annual
West Coast Engineering Management Conference, May 2728, 1957, Los Angeles,
California, sponsored by the Management Division, Southern California Section, The
American Society of Mechanical Engineers.
DETERMINING THE BEST POSSIBLE INVENTORY LEVELS
265
DEMAND
REPLENISHMENT
Figure 5. Branch Warehouse Supply System
Let us look again at Fig. 5, which shows the relation between the central
and branch warehouses. With the oneforone replenishment system which
is being used, the number of sales during any week of a part at a branch
(and the number of lost sales) depends only upon the sales demand during
that week and the inventory on hand at the start of the week. This starting
inventory depends, in turn, upon sales during previous weeks, the lengths
of time required for replenishment cycles during preceding weeks, and the
level of inventory which was initially established.
Thus, in order to determine the actual sales and lost sales which can be
expected to result from a given initial inventory level, it is necessary to
know only the patterns of sales demand and replenishment cycle times
which occur. Both demand and replenishment times are uncertain, but
typical patterns for them can be constructed from the frequency distribu
tions which describe them. Conceptually, we can think of generating these
patterns by each week spinning the two roulette wheels shown in Fig. 5.
Since actually doing this would require too much of our own time, we let
the electronic computer calculate the values which result from spinning the
roulette wheels. A highspeed computer can run through calculations of this
type very, very quickly, and in a brief time it is possible to determine the
average number of lost sales which will occur each week for any particular
initial inventory level.
Using the computer to simulate history resulted in a set of curves similar
to the one shown in Fig. 6. For a given average weekly sales demand, the
percentage of sales which will be lost decreases as the inventory level in
266
INVENTORY CONTROL
AVERAGE
PERCENTAGE
OF
14% •
12% ■
LOST SALES
10%
PER WEEK
8% ■
6%
4%
2% ■
20 40 60 80 100
BRANCH WAREHOUSE INVENTORY LEVEL
Figure 6. Expected Weekly Lost Sales as a Function of Branch
Inventory Level, for Average Demand of Ten per Week
creases. However, as we see in Fig. 7, the exact relation between the aver
age percentage of lost sales per week and branch inventory level depends
upon the average weekly demand.
THE SOBIL (SIMULATED OPTIMAL BRANCH
INVENTORY LEVELS) SYSTEM
Once we know the relation between average lost sales and inventory
levels, we are ready to use this knowledge to establish the best possible
branch warehouse inventory levels. By "best possible" inventory levels, we
mean those levels of inventory which will result in the largest total expected
profits from the operations of the branches. By balancing the expected gain
resulting from the sales that branch warehouse inventories will support
against the expected cost of carrying this inventory, normal inventory levels
can be established for every part at each branch in a way which leads to
the largest possible total profits. For convenience, we have invented the
name "SOBIL System" to refer to the use of the simulation curves of Fig.
10 for establishing branch warehouse inventory levels which are optimal
in accordance with the expected profit maximization criterion.
The gross gain each week from holding inventory is the product of the
average weekly sales that will result from this inventory and the gross
margin per unit sold, where the gross margin on a part is the difference
between its selling price and material cost. The reasonableness of this can
be seen from the following three considerations.
First, lost sales, as measured by the simulation curves of Fig. 7, are
lost business. This is substantially true, for customers are usually unwilling
to wait for a part when a branch is outofstock. Of course, for very minor
delays of a day or two pending receipt of a shipment already in transit,
DETERMINING THE BEST POSSIBLE INVENTORY LEVELS
267
AVERAGE 20% H
PERCENTAGE OF
LOST SALES
PER WEEK
15%
10%
5%
0%M»
® \® \® \(5
O INDICATES AVERAGE
WEEKLY DEMAND
45
60
— i—
90
105 120 135 150
BRANCH WAREHOUSE INVENTORY LEVEL
Figure 7. Expected Weekly Loss Sales as a Function of Branch
Inventory Level, for Various Average Weekly Demands
some customers may be willing to wait, but this possibility has in fact
already been taken into account by the manner in which the simulation
curves were derived.
Second, in those rare instances when a customer who requests a part
from a branch which is out of stock is willing to experience a substantial
delay in obtaining it, i.e., to allow his order to be backlogged, there are
considerable extra costs incurred in processing, expediting, handling, and
shipping this order. These extra costs probably eliminate profit which
would have accompanied a normal sale of the item.
Third, the administrative and processing costs of handling customer
orders are the same regardless of whether those customer orders result in
direct sales or lost sales.
On the opposite side of the ledger, the penalty attached to holding inven
tory is the cost of carrying this inventory. Specifically, we must consider
those variable elements of inventory carrying costs which depend upon the
size of inventories actually held, as e.g., the cost of invested capital, obso
lescence, insurance and taxes, handling and labor costs, and space costs.
In Replacement Division, the variable costs of holding inventory at a branch
warehouse are proportional to the value of that inventory, where inventory
is valued at material cost per unit times the number of units.
268
INVENTORY CONTROL
We can define the net gain from holding inventory to be the gross gain
resulting from this inventory less the variable costs of holding this inventory.
In terms of our analysis, the weekly net gain from an inventory of a part at
a branch is simply the expected weekly sales of the part that this inventory
level will support times the gross margin per unit, less the weekly variable
cost of carrying this inventory. The optimal inventory level for a part at a
branch, as defined by the SOBIL System, is the amount of inventory which
will maximize the weekly net gain. If all branch warehouse inventory levels
were established according to the SOBIL System, then the overall expected
profits of the operation will be maximized.
D = average weekly demand for an item at a branch warehouse
X = branch warehouse inventory level for the item
/(X) = average weekly lost sales for the item at the branch, as a percentage of the
demand determined by Monte Carlo approach
P = selling price of the item
C = material cost of the item
v — weekly variable inventory carrying costs, as a percentage of branch ware
house inventory value
G(X) = weekly net gain from a branch warehouse inventory level of X for the item
Criterion: the optimal branch warehouse inventory level for the item is that X that
maximizes G(X). This value of X must satisfy the inequality
G(X 1) <G(X) >G(X+1),
where
G(X) = (P  C)[l  /(X)]D  vCX.
The inequality is equivalent to
(PC)D/(X1) vC> (PC)D/(X) < (PC)D/(X+ 1) + vC.
The value of X that satisfies this inequality can be closely approximated by solving
the equation obtained by setting the derivative of G(X) with respect to X equal to
zero. This leads to
d/(X) vC
dX (PC)D
This means that the solution is equal (or very nearly equal) to the value of X where
the negative slope of the /(X) curve is equal to vC r (P — C)D.
Figure 8. The SOBIL System
The procedure for establishing the best possible inventory level is shown
symbolically in Fig. 8. The weekly net gain from holding an inventory of
a part at a branch is a function of the size of that inventory. In order to find
the optimal size of that inventory, we can set the derivative of the weekly
net gain from holding inventory equal to zero, and then solve this equation
for the optimal inventory level. When we do this, it turns out that the best
possible inventory levels do not depend upon the height of the simulation
curves of Fig. 7, but, rather, they depend upon the slopes of these curves.
DETERMINING THE BEST POSSIBLE INVENTORY LEVELS
269
In particular, in order to establish the optimal inventory levels, we have to
know the relation between the rate of increase in average weekly sales and
branch warehouse inventory levels. The form of this relation is shown in
Fig. 9, where we see that for any given level of average weekly demand,
the rate of increase in average weekly sales decreases as the inventory level
increases. The electronic computer can be used to produce these new curves
shown in Fig. 9, as well as the simulation curves shown in Fig. 7.
.50
RATE OF
INCREASE IN
AVERAGE .40
WEEKLY SALES
30
20
.10
.00 1 "
Q INDICATES AVERAGE
WEEKLY DEMAND
45 60 75 90 105 120 135 150
BRANCH WAREHOUSE INVENTORY LEVEL
Figure 9. Rate of Increase in Expected Weekly Sales as a Function
of Branch Inventory Level, for Various Average Weekly Demands
To determine the optimal inventory level for a part of a branch, we first
select the curve from Fig. 9 which corresponds to the average weekly
sales demand for the item. Suppose that this particular curve is the one
shown in Fig. 10. Next, we divide the variable cost of carrying one unit of
inventory for a week by the gross margin on the part. The resulting value,
as we see in Fig. 10, is found on the vertical axis. From there, we move
horizontally from the curve, and then we move vertically from the curve to
the horizontal axis. The level of inventory which is then indicated on the
horizontal axis is the best possible inventory level for the part at the branch.
If desired, rather than using this graphical procedure, the determination
of optimal branch inventory levels can readily be programmed for an
electronic computer.
It must be emphasized that while the SOBIL System provides an auto
matic way of determining branch warehouse inventory levels, this neither
restricts management's prerogative nor eliminates the need for sound busi
ness judgment. Management always has the discretion to establish branch
270
INVENTORY CONTROL
RATE OF INCREASE
IN AVERAGE
WEEKLY SALES
(D) de(X)
dX
vC
PC
OPTIMAL X
BRANCH WAREHOUSE
INVENTORY LEVEL X
Figure 10. SOBIL System, Graphical Determination of Optimal
Branch Inventory Levels
warehouse inventory levels other than those indicated by the SOBIL System,
if policy considerations should so dictate. Indeed, management might want
to do so because of a desire to give exceptional service to favored customers,
to maintain a particular shareofthemarket, or aggressively to develop a
new market. In such cases it is easy to determine the net gain from holding
inventory which is foregone by establishing a branch warehouse inventory
at some level other than that indicated by the SOBIL System. This provides
additional information which management would not otherwise have, in
formation which can help management decide whether the special policy
considerations are worthwhile. Figure 1 1 shows the form in which this in
formation could easily be provided if the calculations of optimal branch
warehouse inventory levels are done on an electronic computer.
forecasting requirements for the
sobil system
In order to make effective use of the SOBIL System, it is necessary to
forecast the average weekly demand for every item at each branch. This
may not be easy to do, for these average weekly demands slowly change
over time.
Over a long period, the local demand for an automotive replacement part
will vary because of the changing size and age distribution of the existing
DETERMINING THE "BEST POSSIBLE" INVENTORY LEVELS 271
stock of automobiles and trucks, the changing number of models in which
the part has been used, competition, seasonal factors, and random fluctua
tions.
Many alternative methods for forecasting demand by item by branch
should be investigated. Perhaps the simplest procedure is to use a moving
average or linear trend extrapolation based on recently experienced demand.
More sophisticated would be the use of a life cycle growth and decay curve
based on the number of models in which a part has been used and the
elapsed times since the part was first and last incorporated in a new model.
Even more elaborate would be a mortality, agedistribution, shareofthe
market model. The size of the market for a replacement part would be esti
mated by considering the size and age distribution of the existing stock of
automobiles and trucks, the probability of the part becoming defective as a
function of the age of the machine in which it is housed, and the models in
which the part has been incorporated. The size of the market for the part
is then multiplied by each branch warehouse's assumed shareofthemarket
to yield the forecasts of the average demand for that part at the various
branch warehouses.
Any or all of these forecasting procedures might be improved by making
adjustments for seasonal variation.
How often the forecasts of average weekly demand should be revised
depends upon how frequently the branch warehouse inventory levels should
be changed. This, in turn, depends upon the clerical and dataprocessing
costs required to reset the branch inventories, the net gain from holding
inventories which is foregone because of using incorrect estimates of average
weekly demand, and any changes which might occur in the values of critical
parameters, such as the means or variances of the replenishment cycle
times, the cost of capital, or other inventory holding costs.
TESTING THE SOBIL SYSTEM
Before completely accepting the worth whileness of the SOBIL System,
it is desirable to estimate the magnitude of the increased profits which
should result from the use of this procedure. There are two general ways of
making such an estimate, either through a careful analysis of historical data
or through a program of controlled experimentation.
The cheapest and fastest way of estimating the amount of increased
profits which should result from adopting the SOBIL System is to assume
that the equation defining the net gain from holding inventories and the
simulation curves expressing the dependence of average lost sales on branch
warehouse inventory levels are all accurate. On this basis, it is possible to
compare the net gain from holding inventories which was actually experi
272 INVENTORY CONTROL
enced with the net gain from holding inventories which would have occurred
had the branch warehouse inventories been established at the levels indi
cated by the SOBIL System.
Using historical data on actual sales, lost sales, and branch warehouse
inventory levels, it is possible to compute what the net gain from holding
inventories actually was during some past period. This can be done for a
randomly selected sample of items and branches, or, if time and budget
permit, for all items and branches.
From the same historical data on actual demand, it is possible to deter
mine the potential performance of the system, i.e., the net gain from holding
inventories which would have resulted from optimal inventory levels. The
results of these two calculations should then be compared. The difference
between the net gain from holding inventories had optimal branch ware
house inventory levels been established and the actually experienced net gain
indicates the increased profits which should results from using the SOBIL
System.
The conclusive test of the SOBIL System can come only from a con
trolled experiment, however. Using proper experimental design principles,
a randomized sample of branch warehouses should have their inventory
levels established as indicated by the SOBIL System. In the remaining
branches, the inventory levels should continue to be determined by the
present system. The worthwhileness of the SOBIL System can then be
determined by comparing the profitability of those branches where it was
employed with the remaining branches. In order to get a meaningful basis
of comparison which is independent of size, what should be considered is
the percentage change in profits of the branch warehouses, not the absolute
profits themselves. Then, the measure of effectiveness of the SOBIL System
would be the ratio of the percentage change in profits of those branches
using this new system to the percentage change in profits of those branches
using the present method.
CONCLUSION
In this article we have described a procedure, which we have called the
SOBIL System, for establishing the best possible branch inventory levels in
a network composed of one central and several branch warehouses using a
oneforone replenishment system. Since the branch warehouse inventories
function as buffers mediating the uncertainties of weekly sales, demand and
replenishment cycle time, there is a probabilistic dependence of weekly lost
sales on branch inventory levels. A Monte Carlo approach, that is, a simu
lation of the history of the system on an electronic computer, was used to
determine the relation between average weekly lost sales and inventory
levels. By balancing the expected gain resulting from the average weekly
sales that branch warehouse inventories will support against the expected
DETERMINING THE BEST POSSIBLE INVENTORY LEVELS
273
cost of carrying inventories, our knowledge of the relation between average
weekly lost sales and inventory levels can be used to determine optimal
branch warehouse inventories, i.e., the levels of inventory which yield the
longest possible net gains from holding inventory.
In addition to outlining the conceptual framework of the SOBIL System,
we have indicated graphical and computational techniques which could be
used in implementing the procedure, discussed the nature of the forecasts
which must be made, and presented ways of determining the amount of
increased profits which should result from adopting this system.
In a warehouse network such as we have described, adoption of the
SOBIL System should result in several advantages. First and foremost, the
total profits of the business should be increased, because of the optimal
Part P
Branch Warehouse B
Average Weekly Demand 10
Branch
A verage
Net Gain
Foregone Net
Warehouse
Percentage
from Holding
Gain from Holding
Inventory Level
of Lost Sales
Inventory
Inventory
40
20.0
6.40
1.10
45
11.5
7.05
.45
50
6.5
7.35
.15
55
3.0
7.50
Optimal Branch
Inventory Level
60
1.7
7.43
.07
65
.8
7.32
.18
Figure 11. SOBIL System, Computer PrintOut
balancing of revenues and costs generated by inventory. Furthermore, for
any given amount of capital invested in inventory, the best possible distri
bution of this inventory can be obtained between the branches and for the
different items, and the total investment in inventory can be controlled
merely by changing the cost of capital. Finally, when policy considerations
dictate establishing branch warehouse inventories at levels other than those
indicated as optimal by the SOBIL System, the amount of shortrun profit
which is foregone by this policy is readily calculable.
acknowledgments
This article discusses some continuing research which is being done for
the Replacement Division of Thompson Products by the Management
Sciences Department of The RamoWooldridge Corporation. This work
represents a team effort, and it is difficult adequately to delineate the parts
for which various people are responsible. The members of the Ramo
Wooldridge team who, at various times, were involved in this project in
274
INVENTORY CONTROL
elude Mr. W. R. Hydeman, Dr. William Karush, Mr. L. A. Moody, Mr.
A. F. Moravec, Dr. A. Vazsonyi, Mr. Jack K. Weinstock, and Dr. David
M. Young. The author's own contributions, conceived in close collabora
tion with Mr. Moravec, were mainly connected with developing the decision
rules for determining optimal branch warehouse inventory levels (based on
the relation between expected lost sales and inventory) and the general
economic analysis of the Monte Carlo model's relevance to the Replacement
Division's operations.
The members of the Management Sciences Department's study team are
especially grateful to Mr. Kenneth C. Lucas of Thompson Products' Re
placement Division for his patience, advice, and cooperation during this
project. The author wishes to thank Messrs. Hydeman and Moravec for
their personal help and encouragement while the three of us were in Cleve
land working on the implementation of the Monte Carlo model.
EDITORS' NOTE
In the preceding article, two important pieces of information, weekly
sales and replenishment cycle time, were obtained by a simulation tech
nique called Monte Carlo.
To illustrate how these data were actually obtained, we can start with
Figure 2 from the article. In the diagram below, assumed values for the
frequency have been inserted on the vertical axis.
9 10 11 12
WEEKLY DEMAND
20
Figure 1
DETERMINING THE BEST POSSIBLE INVENTORY LEVELS
275
From the frequency distribution of weekly demand, a probability func
tion is easily derived. The frequency distribution represents 184 weeks of
experience (the summation of all the vertical bars). Since a weekly demand
of 2 units occurred twice during the 1 84 weeks, we can say that the proba
bility of a demand of 2 units in any given week is 2 / 184 or slightly over 0.01.
Similarly, the probability of a demand of 10 units is
23/
184
or 0.125. When
these probabilities are plotted against the possible values of weekly demand,
we get the discrete (since we are dealing with whole demand units only,
excluding demands like 4.7 units) probability function shown below in
Figure 2.
0.12
0.11
0.10
0.09
sr 0.08
Z 0.07
u 0.06
u
O
O 0.05
0.04
I 003
0.02
0.01
8 9 10 11 12 13
WEEKLY DEMAND  D
14 15
Figure 2
The job of simulating demand by the Monte Carlo technique can be
made somewhat easier if this probability function is transformed into a
"cumulative" probability function. Whereas the probability function in
Figure 2 shows the probability that weekly demand will equal a certain
number of units, a cumulative probability function shows the probability
that weekly demand will be equal to or less than a given amount. As an
example, in Figure 2 the probability of a demand of 4 units per week is
0.0326. The probability of a demand of 4 units or less would be the sum
of the probability of a demand of units, plus the probability of a demand
of 1 unit (which can be written P(D — 1) where D denotes demand) plus
276
INVENTORY CONTROL
8 9 10 11 12 13
WEEKLY DEMAND D
Figure 3
14 15 16
19 20
P(D = 2) plus P(D = 3) plus P(D = 4) or + + 0.011 +0.022 +
0.033 = 0.066. It should be clear that the cumulative probability for the
highest demand shown on Figure 2 (i.e., 19 units) is 1.0, since demand
in every week of the 184 weeks shown in Figure 1 was 19 units or less,
P(D ^ 19) = 1S4 /is4 = 1.0. Figure 3 is the cumulative probability func
tion plotted from Figure 2. Table I presents the same information in
tabular form.
This cumulative probability function is the basis for simulating demand
by the Monte Carlo method.
Suppose now we determine a three digit random number. Possible sources
of random numbers would include a computer routine which generates
random numbers such as the one used in this article, a random number
table, drawing one number from each of three jars containing balls num
bered through 9, or throwing a ten sided die three times. There are 1000
possible numbers that may show up on the draw of a three digit random
DETERMINING THE "BEST POSSIBLE" INVENTORY LEVELS 277
TABLE I
Probability That
Weekly Demand Is
Demand
Less than or Equal
D
to D
1
2
0.011
3
0.033
4
0.066
5
0.109
6
0.174
7
0.261
8
0.359
9
0.467
10
0.592
11
0.696
12
0.783
13
0.848
14
0.891
15
0.924
16
0.951
17
0.973
18
0.989
19
1.000
number — 000 through 999. The probability of drawing any given number
is 1/1000 and is the same for all possible numbers. That is, P (Number ==
000) = P(Number = 001) = . . . = P(Number — 999). Looking back
at Table I, we see that the desired probability for a weekly demand of 2
units or less is 0.011. This demand can be simulated by saying that every
time a random number which is between 001 and 011 is drawn we will
automatically set weekly demand equal to 2 units. Carrying on in this vein,
if the random number is 012 to 033, we will set demand equal to 3 units.
Similarly, if the random number is 784 to 848, demand equals 13 units;
and demand equals 19 units for random numbers from 990 to 000. (We
interpret 000 as the number 1000 rather than 0.)
Looking back at this procedure, let us check the results with Figure 2.
The probability of a random number falling between 001 and 011 is
Hloeo or 0011, which, according to Figure 2, is exactly the probability
that demand equals 2 units. The probability of a random number falling
between 990 and 000 is also Hiooo* which is exactly the probability of
demand equal to 19 units per week. Thus, we have a technique which
enables us to simulate exactly the distribution or probability function of
weekly demand.
One question still remains. The Monte Carlo method insures that de
mands occur with the same relative frequencies as shown in Figure 2.
Using a simple average demand, however, would also yield the same total
demand over the 184 week period. Why, then, bother with Monte Carlo
278 INVENTORY CONTROL
simulation? The important difference between such a straightforward repro
duction of total or average demand and the Monte Carlo technique is that
Monte Carlo allows a randomized arrangement of individual weekly de
mands. Thus, unlikely, but nonetheless possible, combinations of demand
like 2 units in one week and 19 units the next will pop up occasionally in
the Monte Carlo method. When such fluctuations arise, the response of the
inventory system can be observed. Simulating demand at a single, average
rate would never allow the testing of the inventory rules under fluctuations
in demand that are certain to exist in the real world. In this article, varia
tions in demand are very important since the magnitude of the inventory
level is specified to protect against such variations.
The replenishment cycle time, which was also simulated by the Monte
Carlo method, could be developed by using a procedure very much like the
one developed for simulating weekly demand.
Chapter 1 4
STATISTICAL METHODS
IN INVENTORY CONTROL
XVI.
Physical Inventory Using
Sampling Methods*
Marion R. Brysont
INTRODUCTION
Business, industry, and government are all faced with the task of periodi
cally taking a physical inventory of all goods on hand. The most commonly
used method of performing this physical inventory is to make a complete
count of all stocks once each period, usually annually. All counts of stocks
are made within a short period of time, say within a week. At the close of
such an inventory, it is assumed that accurate knowledge of the quantity
and value of all stocks on hand has been obtained. Unfortunately, in most
cases this may be a false assumption.
In most of the larger and many of the smaller establishments a con
tinuous record is kept of stocks. These records are altered as activity
* From Applied Statistics, Vol. 9, No. 3 (1960), 178188. Reprinted by permission
of Applied Statistics.
t The author wishes to express his thanks to Professor R. F. Rinehart, who critically
read the manuscript and made many helpful suggestions.
279
280 INVENTORY CONTROL
changes the quantity on hand. If the records were completely accurate it
would, obviously, be unnecessary ever to make a physical count of stocks.
Many of the records are inaccurate for the following reasons: the previous
inventory was in error causing the record to reflect the wrong balance from
the beginning of the fiscal period; some receipts and/or issues were made
without correct alteration of the records; items were lost; items were pil
fered; records were lost; and other record adjustments, such as price changes
or reserve stock levels, were incorrectly processed. Hence it is the purpose
of the inventory to correct the records which reflect wrong information.
The question which immediately arises is 'How accurate are the records
at the close of the inventory?' The personnel employed in the inventory are
relatively untrained in inventory methods since this activity occurs during
only one week out of the year. During this week personnel are drawn from
all parts of the establishment to perform the counting and clerical duties
incidental to the inventory. Many of these people are not interested in in
ventory work, and if they are doing satisfactory work at their normal job
they know they will not be dismissed or demoted if their poor inventory
work is detected.
Because of the suspected inaccuracies in inventory and because opera
tional record errors occur and remain undetected for as long as one year,
an agency of the U.S. Government sponsored a research project to deter
mine what could be done about improving inventory methods.
DESCRIPTION OF SUPPLY CENTERS
The government agency mentioned above maintains supply centers in
various parts of the country. Each of these supply centers has in its storage
areas from 5,000 to 150,000 different types of item stored. Hereafter the
term 'item' will be used to denote a type of item and the term 'piece' to
denote an individual part. From zero to more than one million pieces of
each item may be on hand at any given time. The storage areas are in the
nature of huge stockrooms whose function it is to receive and store in
coming goods and issue and ship goods ordered by their customers. The
supply center maintains records of stocks on hand and the records depart
ment receives orders for shipments, processes them, and instructs the stock
room to issue the stock. It also processes all receipt vouchers and instructs
the stockroom to store the stock in an assigned location.
Each item has an assigned location or locations in the stockrooms. When
an order comes in to the records department, the records are checked to
determine whether the stock in this quantity is available for issue. If it is
not, the order is sent to a master records centre which will reorder the stock
from another storage center. If the stock is available, a shipping order is
PHYSICAL INVENTORY USING SAMPLING METHODS 281
drawn up and sent to the stockroom. This shipping order designates the
item number, location, and quantity to be shipped, among other facts. If the
item is stored in more than one location, one of the locations is designated
as the master location. The recorded balances are kept by item number and
not by location; hence the single balance kept by the records department
reflects the sum of the quantities in each storage location. The records
department adjusts its balance to reflect the issue. Receipts are processed
in essentially the same way.
BASIC PLAN OF OPERATION
One such supply center was selected by the government agency as the
site of the study. After nine months of preliminary work at this center, an
additional center was assigned for further study and testing. Each of these
two centers has approximately 100,000 items in its stockrooms.
Various methods of quality control have been used from time to time in
the actual taking of a shutdown inventory. These have been principally
confined to quality control of the individual counter and quality control of
the areas of the stockroom after the counting was complete. These methods
have not proved to be very successful in the inventories taken by the agency
which sponsored this research.
The basic concept of the present project is the continuous quality control
of the storage areas. Those areas which are deemed to be out of control
are completely inventoried. This serves the same purpose as 100% inspec
tion of a rejected lot of pieces produced on a production line.
The stockrooms of a supply center are stratified by physical location.
All lots are of approximately equal size. An attempt is made to store the
more active items together in one or more lots and to store the less active
items in different lots. In addition to the advantage this 'popularity' storage
has of placing the active items nearer the shipping facilities, it also serves to
increase the betweenlot variance of the error rates.
QUALITY CONTROL PROCEDURE
To initiate the quality control once the stratification is complete, a sample
of items in each lot is selected. A physical count of the number of pieces of
each of these items in the stockroom is made and this count is compared
with the recorded balance. If the count disagrees with the balance by more
than the 'leeway,' the item is said to be 'discrepant' or 'in error.' This
leeway is established for the discrepancies so that a minor discrepancy may
be ignored. In this study an item is classed as discrepant if and only if the
difference between the count and the balance is greater than 1% of the
balance, or the monetary value of the discrepancy is greater than $1
(approximately 7s.).
282 INVENTORY CONTROL
Following the count, an estimate of the percentage of items in each lot
which are discrepant is made. The simple binomial estimator
lOOd;
rti
is used. Here
Pi = the estimated percent discrepant in the ith lot; (hereafter
called 'estimated error rate').
d t — number of discrepant items in the sample from the ith lot.
rii = sample size in the zth lot, i.e., the number of items.
The variance of this estimator is the usual one :
P 4 (100f\)
var Pi —
rii
Next, an acceptable limit for the estimated error rate is established. In
this project the limit is 10%.
Hereafter the words 'sampling' or 'sample' refer to the process of taking
a physical count on only a sample of items in a lot. The word 'inventory'
will refer to a complete count of every item in the lot and the reconciliation
of this count with its recorded balance.
The results of the sample are then observed. If more lots have an esti
mated error rate of above 10% than can be inventoried by the counting
crew in a threemonth period (quarter), only enough of the highest error
rate lots are rejected to consume one quarter of inventory time. The re
mainder of those above 10% are temporarily accepted.
In the first quarter of the year, those lots rejected on the initial sample
are inventoried. Those lots which were accepted, whether or not their
estimated error rates were above 10%, are resampled as before during this
quarter. From this group of resampled lots the inventory load for the
second quarter is chosen. It will be seen that those lots inventoried in the
first quarter cannot be inventoried again during the second quarter since
they were not included in the resampled lots during the first quarter. They
will be resampled during the quarter following their inventory, i.e., during
the second quarter in this case, so cannot be rescheduled for inventory
before the third quarter.
If the lots whose estimated error rates are above 10% do not constitute
a full quarter of inventory work, either the acceptable limit can be lowered
or the size of the inventory crew can be reduced.
PHYSICAL INVENTORY USING SAMPLING METHODS 283
OPERATION OF INITIAL STUDY
In October of 1958 the project was initiated at supply center number
one (SC1). For the first nine months the work was confined to methods
of stratification and methods of counting an item without freezing activity
on the item. A team of 15 men was assigned to work on the project. The
center was stratified into 25 lots and two samples were taken in each lot,
the time between samples being about four months.
A completely random sample of items in each lot was drawn. First, for
sampling purposes, every piece of a given item was considered as being
stored in the lot with the master location of the item. A cumulative list of
all possible storage locations was drawn up. This list is not the same as that
kept in the 'location file' from which issues are drawn up. For example, a
given rack may have as many as three items stored on it, all of which have
the same numbered location in the location file. This rack was considered
as three 'item locations' even though the records showed it as only one
location. It was because of this that it was not possible to sample from the
location file. This would result in clusters of items in the sample, the cluster
size varying from one to twentyfour items.
The sample size was not rigidly fixed in advance. A sample of item loca
tions was drawn from the cumulative list, using random number tables, and
each of these locations was visited. If the location was the master location
of some item, this item was included in the sample. If it was a reserve loca
tion or an empty location to which no stock number was assigned, the
location was dropped from the sample. If the final sample size was too
small, a second sample was taken to augment the first one. A sample size of
between 200 and 300 was selected in all but a few small lots. The average
lot size was about 4000.
The count on the sample items was taken without affecting the normal
center operation. In simplified form the steps followed were:
(1) Count all pieces of an item in the stockroom.
(2) Obtain the recorded balance for the item.
(3) Compare the two figures.
(4) If the figures agree, classify it as a nondiscrepant item.
(5) If the figures disagree, they may disagree because
(a) A miscount has occurred.
(b) An activity has occurred in the record section but has not yet
occurred in the stockroom.
(c) The item is discrepant.
This item is set aside for two weeks.
284 INVENTORY CONTROL
(6) Two weeks after the first count, retrace steps 14 if the item is
in disagreement.
(7) If the two figures now agree, it is a nondiscrepant item.
(8) If they disagree by the same quantity as they did on the first count,
the item is classed as a discrepant item. Points (a) and (b) in step
5 are improbable in this case since the same miscount is unlikely
and no activity should be in process for two weeks.
(9) If the item disagrees by an amount different from the first discrep
ancy, set it aside for another two weeks.
(10) A third count is taken on all items falling into step 9 and steps
14 and 8 are followed.
(11) For items which are still not reconciled, an investigation is con
ducted to determine their true nature.
Several minor refinements can be and have been made in the above pro
cedure but basically it has been operated successfully using this system.
The purpose of the second sampling was to estimate the error rates
which existed in each of the 25 lots. The error rates obtained in this sample
were then compared with the results of the annual shutdown inventory
which was taken immediately following the second sampling phase.
RESULTS OF THE INITIAL STUDY
Table I gives the lot sizes, sample sizes, and the error rates of both the
sample and the shutdown inventory.
From the table it will be noted that the difference between the sample
results and the findings of the inventory is significant at the 95 % level for
more than half of the lots. Since it was the purpose of both the sample and
the inventory to estimate the true error rate of the lot it appears that some
thing is wrong. It could be one or both of two things:
(1) the sample was drawn incorrectly, biasing the results;
(2) the sample does not estimate the percentage found by the inventory
but some other quantity; either or both may differ from the true
error rate.
Let us look at the first of these two possibilities. In order to check the
accuracy of the sampling, the inventory findings on the sample items only
were compared with the inventory findings on the entire lot. The difference
between these two figures represents sampling error only. Table II gives the
results of this comparison.
In this comparison only three of the twentyfive lots had differences
which were significant at the 95% level. This leads one to believe that the
sampling was without bias with the possible exception of lots 1 and 10.
Since one of these is negative and the other is positive, and since there are
PHYSICAL INVENTORY USING SAMPLING METHODS
285
TABLE I. ERROR RATES (%)
Estimated
Shutdown
Sample
Error Rate
Inventory
Difference
Lot No.
Lot Size
Size
from Sample
Error Rate
(Inv.Samp.)
1
7,818
310
61.9
52.3
 9.6t
2
7,191
291
57.7
53.0
 4.7
3
1,809
217
51.6
54.8
3.2
4
2,344
66
46.9
50.9
4.0
5
6,717
285
43.5
50.5
7.0*
6
3,228
220
41.8
32.9
 3.9*
7
7,945
300
39.0
41.5
2.5
8
3,493
212
37.3
47.0
9.7*
9
3,676
207
37.2
31.2
 6.0
10
3,136
217
36.9
53.7
16.8t
11
1,908
207
35.3
33.3
 2.0
12
5,053
259
34.8
50.5
15.7t
13
7,271
287
33.1
47.5
14.4t
14
2,930
213
31.9
34.3
2.4
15
6,920
239
31.4
37.5
6.1*
16
2,885
234
29.5
41.0
11.5t
17
614
72
29.2
39.9
10.7*
18
3,004
222
27.5
37.4
9.9t
19
7,004
233
27.0
25.7
 1.3
20
1,918
226
26.5
31.2
4.7
21
2,266
197
24.4
28.6
4.2
22
3,175
216
22.2
33.4
11.2t
23
6,248
260
21.9
25.4
3.5
24
1,491
192
20.3
25.5
5.2
25
2,872
242
16.5
25.6
9.1t
Total
192,376
5,624
_
Mean
—
—
34.7
40.7
6.0
i
* Difference significant at 95% confidence level.
f Difference significant at 99% confidence level.
equal numbers of plus and minus differences, no consistent bias is evident.
Let us look now at the second possible cause of the large differences
shown in Table I.
Since the differences in error rates given by Inventory (Inventory denotes
this particular inventory) and the estimates given by the sampling plan
(hereafter referred to as SQC) are not due primarily to the sampling tech
nique of SQC, these differences must be due in large part to mechanical
mistakes by SQC, or mistakes by Inventory, or change in discrepancy status
of items from one count to the other. The latter group is regarded as not
significant. It becomes then a question of determining the relative order of
magnitude of the mistakes made by SQC and those made by Inventory.
Information on the mistakes of SQC and those of Inventory is provided
through a study of the items counted by both SQC and Inventory. The
results of this study follow:
286 INVENTORY CONTROL
1. Inventory and SQC agree on a total of 81 percent of the items, the
nondiscrepant items in agreement being 58 percent, and the discrepant
items 23 percent.
2. Inventory found a discrepancy, SQC found no discrepancy, on
7.75 percent of the items. In a further study it was found that Inventory
mistakes account for 7 percent of these, and the other 0.75 percent are
actual discrepancies which occurred after the sampling count was com
pleted.
3. SQC found a discrepancy, Inventory found no discrepancy, on
approximately 15 percent of the items. Essentially all of these differences
result from SQC mistakes.
4. SQC and Inventory both found a discrepancy, but disagreed about
its magnitude, on 9.75 percent of the items. By special investigation on
these items, it was found that SQC and Inventory shared about equally
in the mistakes, but that about 2% of the mistakes were due to an actual
change in the size of the discrepancy between the time of the sample
count and the time of the inventory count.
TABLE II. ERROR RATES (%)
Lot No.
Inventory Error Rate
Inventory Error Rate
Difference
on Sample Items
on A 11 Items in Lot
{LotSample)
1
61.3
52.3
 9.0t
2
55.5
53.0
 2.5
3
57.8
54.8
 3.0
4
47.8
50.9
3.1
5
50.7
50.5
 0.2
6
39.3
32.9
 6.4
7
46.2
41.5
 4.7
8
45.5
47.0
1.5
9
41.4
31.2
10.2*
10
43.3
53.7
10.4t
11
39.2
33.3
 6.2
12
46.5
50.5
3.9
13
43.8
47.5
3.7
14
32.7
34.3
1.6
15
35.5
37.5
2.0
16
40.6
41.0
0.4
17
40.6
41.0
0.4
18
34.7
37.4
2.7
19
29.3
25.7
 3.6
20
31.3
31.2
 0.1
21
28.6
28.6
0.0
22
29.2
33.4
4.2
23
30.4
25.4
 5.0
24
26.8
25.5
 1.3
25
22.4
25.6
3.2
Mean
39.9
40.7
0.8
* Difference significant at 95% confidence level.
t Difference significant at 99% confidence level.
PHYSICAL INVENTORY USING SAMPLING METHODS 287
From this analysis we arrive at the figures in Table III. In accordance
with this table, for example, the SQC technique results in 22 mistakes for
every 1000 good items counted, and in 122 mistakes for every 1000 dis
crepant items counted. The rate of mistakes in general is 5.5 percent of the
time for SQC and 1 1 percent of the time for Inventory. The general con
clusion is that overall the SQC technique makes only half as many mistakes
as the Inventory technique.
TABLE III. FREQUENCY OF MISTAKES,
COMPARED WITH SQC
INVENTORY
Nondiscrepant Discrepant
Items Items
All
Items
SQC 2.2% 12.2%
Inventory 10.4% 12.2%
5.5%
11.0%
This explains the differences shown in Table I. Of the thirteen significant
differences shown there, only two of them are negative and in these two
cases the difference is largely explained by sampling bias or sampling
error, as indicated in Table II.
There are two major reasons for the superior performance of the SQC
team. They are:
( 1 ) The procedure of accepting no count unless it agrees with a previous
count or with the recorded balance virtually eliminates the possi
bility of a miscount.
(2) The team is experienced in inventory and interested in the work.
Results of Second Study
After the first study at SC1, the project was initiated at a second supply
center (SC2). It was also expanded to include the physical inventory of
the rejected lots. In these two centers, no shutdown inventory is being
taken.
After some study it was decided that a center should be divided into
sixteen lots of equal size. If more than three lots had an estimated error
race of more than 10%, the three lots showing the highest percentage error
in any quarter would be completely inventoried the following quarter. The
principal advantage of this is that the work load in each quarter is the same
so that it is not then necessary to have a variablesized inventory team.
In any given quarter thirteen lots are sampled and three lots are inven
toried so that each year an equivalent of threequarters of the lots in the
entire center is inventoried completely. Actually some lots may be inven
toried twice in a year so that more than onequarter of the centre may miss
inventory entirely. Adjustments in the recorded balances for discrepant
items are made.
It might at first sight be thought paradoxical that the recommended plan
288 INVENTORY CONTROL
should include a proportion of complete inventory work, since the earlier
results indicated that more mistakes were made in a complete inventory
than in a sample count. However, it must be stressed that the earlier com
parison was between an experienced SQC team and relatively untrained
personnel carrying out the inventory. A team specialising in inventory work
would be expected to have a low mistake rate. Furthermore, the periodic
complete inventory allows proper adjustment of erroneous balances to be
made.
Since the purpose of the sampling is to rank the lots from most erroneous
to least erroneous, the question whether recounts were necessary in the
sampling arose. If only one count is taken on the sample items and the
item is classed as discrepant if its count does not agree with its balance,
ignoring the possibility of miscounts and documents in flow, what effect
would this have on the ranking? This question was studied using the data
on the 5624 items in the sample at SC1. Each item was classed as a dis
crepant or a nondiscrepant item on the basis of the first count only and
the lots were ranked. The ranking thus obtained was compared with the
ranking using the recount procedure. The results of this are presented in
Table IV.
TABLE IV. COMPARISON OF RANKINGS OF FIRST COUNT OF
LOTS BY DIFFERENT METHODS
Rank by Rank by Rank by
Single Count Multiple Count Inventory
1
1
4
2
2
3
3
3
1
4
4
5
5
5
6
6
6
18
7
7
10
8
8
9
9
11
17
10
9
19
11
10
2
12
12
7
13
17
12
14
16
11
15
13
8
16
14
15
17
15
13
18
18
14
19
20
20
20
19
12
21
21
21
22
23
25
23
22
16
24
24
24
25
25
23
PHYSICAL INVENTORY USING SAMPLING METHODS
289
Spearman's rank correlation coefficient has the following values:
(1) single count v. multiple count r 8 = 0.984
(2) single count v. inventory r 8 = 0.754
(3) multiple count v. inventory r 8 = 0.775
The single count agrees extremely well with the multiple count and both
are in reasonable agreement with the inventory findings.
On the basis of the foregoing results, it was decided that for the sample
items, only a single count would be taken. This saves considerable time in
the counting. No adjustments are made on the basis of the sample findings.
During the first quarter, SC2 was sampled, without stratification. On
the basis of this sample, the error rate in each section of each stockroom
was estimated. The sections were then grouped together into lots, sections
with similar estimated error rates being put into the same lots. The three
lots with the highest error rates were inventoried, and the remaining lots were
resampled in the following quarter. In the quarter following the inventory,
the three inventoried lots were resampled.
The results of the samples and the inventories at SC2 are given in
Table V.
TABLE V. RESULTS OF STUDY AT SC2
Lot No.
Error Rate
on
Error Rate on
Error Rate on
First Sample
<%)
Inventory (%)
Second Sample ( % )
1
40.0
29.5
5.9
2
39.8
32.6
5.5
3
34.9
30.1
4.9
4
26.7
24.4
5
25.6
30.1
6
24.0
18.8
7
20.6
24.2
8
19.5
30.1
9
19.4
12.0
10
14.3
11.5
11
12.7
18.8
12
12.0
13.2
13
11.8
10.0
14
11.7
10.4
15
11.1
14.3
16
*
♦
Mean
21.6
30.7
15.6
Mean of Lots
4.15
17.5
18.2
No results available.
It will be noted that the inventory in the first three lots reduced the
error rate radically. It had an overall effect of reducing the centerwide
error rate by six percentage points. It will be noted also that the error rate
in the lots which were not inventoried did not grow significantly during
290 INVENTORY CONTROL
the quarter.
There are no significant differences between the sample estimates in
the first and second quarters except for lot 8. In this instance, much
stock was moved into the lot during the quarter. The stock which was
moved in was believed to have a high error rate by the center personnel.
COMPARISON OF INVENTORIES
The direct cost per item of the SQC inventory is about 7080% of
the direct cost of the shutdown inventory. The saving in cost is the
result of more efficient operation of the SQC inventory and the lack of
necessity for training a large group of people for inventory. Other ad
vantages of the SQC inventory are:
(1) No shutdown period is necessary, so that normal operation is
carried on throughout the year.
(2) At any given time the records have a higher degree of accuracy.
(3) A trained team is available for any special inventories which may
be necessary.
(4) Inefficient or errormaking procedures are quickly identified and
eliminated.
(5) An incentive for more efficient centre operation is produced.
(6) State of preservation of stocks is constantly reviewed.
XVII.
Inventory Policy by Control Chart
J. W. Dudley
THE INVENTORYPRODUCTIONSALES
RELATION
The economic advantages of adequate inventory control in relation to sales
and production at a manufacturing plant are not only important in the
continual struggle against competition, but also reasonably selfevident
to most plant managers.
The average plant manager fully appreciates the value of effective in
ventory control and usually believes that he has an active, accurate and
* From Industrial Quality Control, Vol. 16, No. 7 (1960), 47. Reprinted by
permission of Industrial Quality Control.
INVENTORY POLICY BY CONTROL CHART 291
effective policy for denning and controlling production and inventory in
relation to sales. However, when he is asked to objectively define his
policy, or to critically examine the dollar results of his policy to see whether
it has accomplished the desired and defined ends, the existing policy may
prove to be a rule of thumb, or worse, a rule of many thumbs. In fact, he
may be fortunate if any reproducible policy can be discovered in a group
of related products.
FACTORS AFFECTING CHOICE OF
CONTROL METHODS
When we try to determine what is adequate inventory control in a
particular plant for a particular type of product, we find that many
factors affect our selection of an inventory control policy and procedure.
Some of these are:
a) The number of different grades, types or catalogued items produced.
A mail order house with thousands of items probably has several
inventory policies, each applying to similar or homogeneous groups
of items.
b) The approximate demand rate or sales of each item.
c) The relation between clerical costs and the sales price of items in
inventory. Consider the diversity in handling paper clips, jewelry,
and perishable foods.
d) The cost of holding items in stock.
e) The reliability in predicting future sales, changes in design or produc
tion methods, etc. Many items depend on the vagaries of weather,
styles, fads, and the like.
f) Customer relationships.
THE INPROSALES CONTROL CHART
Variations in the foregoing factors may result in several policies of in
ventory control, ranging from simple rules to highly elaborate formulas
requiring computers to handle hundreds or thousands of separate items. In
the writer's opinion, there is a broad middle ground where a simple control
chart technique with low clerical work load can handle the inventory
problem on generally similar and homogeneous items with economical
effectiveness. This technique is easily adaptable to computers on a large
scale, if required.
It is not within the scope of this discussion to describe the fundamental
basis and mathematical background for statistical control charts. Most
companies have been using these charts in many applications for many
292
INVENTORY CONTROL
years. One of the most important provisions in setting up such a chart is
that the subgroups which are used for computing expected variation and
control limits shall consist of items which are considered to be similar, or
as the statistician says, homogeneous. We believe that most plant managers,
even those to whom control charts are relatively new, will not attempt to
impose similar inventory policies on items of material which are funda
mentally different in handling costs, volume required for storage, etc.
Obviously, judgment must be exercised in deciding how broad the coverage
should be in this application of control charts. We believe that a similar
type of judgment would have to be exercised for any method of inventory
control.
Case A
Case B
Case C
No Prediction
Adjustment of
Target Values
On
Values Based on
Set by Manage
Future Sales
Prediction of
ment, Based on
Future Sales
Experience with
X  E.R
Case A or B
Upper Control Limit
Target Average
Lower Control Limit
Items of Product, in Order of Increasing $ Sales/30 Days
Figure 1. Stages in Development of Inventory Control
STEPS IN CONTROL
In this article, this control chart method is described for the following
situations, as illustrated in Figure 1.
Case A
The case where future sales of product cannot be firmly and accurately
predicted and where it may be desired to balance the production of several
items considered to be similar, statistically homogeneous, and made from
common raw materials or using common machines.
INVENTORY POLICY BY CONTROL CHART 293
CaseB
The case of similar items where total sales can be predicted with fair
accuracy.
CaseC
A case similar to Case B, in which top management has a welldefined
policy based on economic lot sizes and past experience for proportioning
inventory, production, and sales.
The study of economic lot sizes for the producer or the consumer has
been discussed in many previous papers and textbooks. 1 The control chart
method described here is useful, regardless of whether it is or is not
desirable to attempt calculations of economic lot size.
In applying this control chart method and deciding whether to use Case
A, B, or C, it is quite possible that the user might make an error in judgment
as to which case is most suitable for the problem at hand, or he may attempt
to include nonsimilar items on a single chart. For example, he may think
that his inventory policy is so well established that he can proceed im
mediately to Case C. If he is too optimistic in this assumption, the com
pleted chart will probably correct him in short order. Also, in a case of
improper selection of nonsimilar items, the chart will tend to be self
correcting.
CONTROL CHART DEVELOPMENT
In developing a control chart technique for coordinating inventory,
production and sales, it is desirable to use dollar values which are combined
into a single factor for chart study. This combined factor may be called
the InProSales factor, abbreviated as IPS.
For the purpose of our discussion, it will be assumed that the proposed
chart will be based on some convenient fixed operating interval, such as a
calendar month, and that production and sales rates will then be corrected
to a 30day basis. The following data are required to compute a value
for IPS:
/ = $ value of inventory physically on hand at end of previous monthly
interval
S = $ value of sales during previous monthly interval, prorated to 30
day basis
P = $ value per 30 days, gross sales value of production, based on rate
of production at end of previous monthly interval
1 See, for example, Hoehing, W. F., "Statistical Inventory Control," Industrial
Quality Control, Vol. XIII, No. 7, Jan. 1957, pp. 713.
294 INVENTORY CONTROL
O = $ value of outstanding orders not shipped at end of previous
monthly interval
Using the above information, the IPS factor is defined as:
IO + P
IPS = —
S+l
It should be noted that the value of unity is added in the denominator
to cover the situation when monthly sales are zero. When sales are zero,
it is still desirable to have a definite value of IPS for the particular item,
because if production is not stopped, IPS may be still larger the following
month, indicating neglect of the control chart warning.
PROCEDURE FOR CASE A
In Case A, where future sales cannot be firmly predicted, the procedure
for preparing the control chart is as follows:
Step 1: Tabulate (/ — O), P and S for the items to be controlled.
Step 2: Calculate IPS for each item.
Step 3 : Retabulate the items in the order of increasing values of sales, S.
Step 4: Using a suitable subgroup size, calculate upper and lower con
trol limits for individual IPS values using the grand average
(X) of IPS and the average Range (R) of subgroups with
standard control chart formulas for "No Standard Given":
UCL = X + E 2 R = X + 2.66/?
LCL — X — EoR = X — 2.66R
with subgroups of two.
Step 5: Plot the control chart using the results of steps 2 and 4.
The control limits are always calculated for individual values of IPS, using
the following values of E 2 , taken from Manual on Quality Control of Ma
terials, American Society for Testing Materials, 1951.
Subgroup
Size
Factor for Individuals
n
E 2
2
2.660
3
1.772
4
1.457
5
1.290
6
1.184
7
1.109
8
1.054
9
1.010
10
0.975
INVENTORY POLICY BY CONTROL CHART 295
Whenever a value of IPS falls above the upper limit, inventory and/or
production is too high in relation to sales, and adjustments should be made
accordingly. Similarly, the converse is true for items where IPS falls below
the lower control limit.
It will be noted that when there are no sales during a given period, the
value of IPS usually becomes relatively very high. Such values of IPS
should not be included in computing averages, ranges and limits for the
control chart. Usually, those values will be found to be far above the
upper control limit when the chart is completed. Nevertheless, these extra
high values act as a valuable red flag to the user of the chart, and should
therefore be identified each time they occur, if there is any hope of future
sales on the item. The value of IPS in such cases becomes the dollar value
of nonmoving inventory on a very unprofitable item! Naturally, the chart
says they don't conform to good inventory policy.
Since Case A is not dependent on a prediction of future sales, manage
ment will normally strive to establish a target or desirable average value
for the IPS factor. A new chart and new limits are computed each month.
Gradual production changes based on a study of IPS values falling outside
of limits will tend to reduce the width of the control chart limits so that
the entire group of similar items approaches a balanced and controlled in
ventory condition, thereby approaching the management policy goal. When
this has been attained, management can then decide whether the average
IPS value should be raised or lowered, depending on several of the factors
listed at the beginning of this article. The monthly control chart will
therefore be a continual guide to management in adjusting production rates
to conform with desired inventory policy.
PROCEDURE FOR CASE B
If management has been operating for some time under Case A, and
finds that it may be possible to predict future sales of the group of similar
items, then the value X on the control chart may be multiplied by the factor:
Total $ Sales during last period
Total $ Sales predicted for next period
since IPS is almost inversely proportional to sales. The control limits are
then shifted accordingly, based on the predicted value for X. This is con
sidered operation under Case B, and obviously should not be continued
unless reasonably accurate sales predictions can again be made for the
next future time period.
296
INVENTORY CONTROL
PROCEDURE FOR CASE C
After operating successfully for a time under Case B for a group of
similar items, management may wish to establish a firm policy covering a
standard central value for IPS and also standard control limits based on
past experience. This would be considered Case C, which represents a very
desirable goal for management. Such standardized control values for one
group of products might be considerably different from controls for an
other group.
It is again emphasized that this InProSales control chart is a flexible
TABLE I.
EXAMPLE OF DATA FOR CASE A
Rank in
order
(IO)
P
S
IPS*
Item
of sales
$
$
$
Value
1
4
9070
8880
13640
1.32
2
27
44640
28770
39680
1.85
3
29
47320
34170
42300
1.93
4
8
43710
40230
23240
3.61
5
15
48340
32300
32010
2.52
6
25
45310
32590
36960
2.11
7
19
51320
35510
34050
2.55
8
3
16180
5030
11540
1.84
9
16
27760
40560
32090
2.13
10
30
48810
31620
47340
1.70
11
28
44230
9910
41620
1.30
12
2
9390
2080
6180
1.86
13
24
24130
30800
36530
1.50
14
10
40110
23810
25310
2.53
15
22
37740
35790
34680
2.12
16
11
48140
44270
28340
3.26
17
6
11890
8930
17080
1.22
18
12
23840
50310
29660
2.47
19
7
42710
34280
22020
3.50
20
21
42440
40330
34670
2.39
21
1
5490
1680
5610
1.28
22
23
26170
34780
36210
1.68
23
26
37800
33700
38740
1.85
24
5
40040
13400
17070
3.13
25
14
23350
25070
30770
1.57
26
9
28720
28490
23770
2.41
27
17
42830
19330
32380
1.92
28
20
20240
31760
34200
1.52
29
13
46310
41260
30140
2.91
30
18
35240
29570
33370
1.94
*/p<:
/_0 + P
5+1
INVENTORY POLICY BY CONTROL CHART 297
tool which in no way interferes with calculation of economic lot size or
other well known production control methods which may already be in
use at the plant. The IPS technique, particularly in the stage described as
Case A, serves as a simple and rational check on other parts of the produc
tion control system. If management is operating either under Case B or
Case C, and a radical change occurs in the overall market for the group of
products, the situation should be resurveyed by again using the methods
described under Case A.
TABLE
II. EXAMPLE OF CALCULATIONS FOR CASE A
Rank in
order
Range
Item
of sales
IPS
R 2
21
1
1.28
12
2
1.86
0.58
8
3
1.84
1
4
1.32
0.52
24
5
3.13
17
6
1.22
1.91
19
7
3.50
4
8
3.61
0.11
26
9
2.41
14
10
2.53
0.12
16
11
3.26
18
12
2.47
0.79
29
13
2.91
25
14
1.57
1.34
5
15
2.52
9
16
2.13
0.39
27
17
1.92
30
18
1.94
0.02
7
19
2.55
28
20
1.52
1.03
20
21
2.39
15
22
2.12
0.27
22
23
1.68
13
24
1.50
0.18
6
25
2.11
23
26
1.85
0.26
2
27
1.85
11
28
1.30
0.55
3
29
1.93
10
30
1.70
0.23
Averages
2.131
0.553
E 2 R
1.471
Upper Control limit, UCL
3.602
Lower Control limit, LCL
0.660
298
INVENTORY CONTROL
EXAMPLE OF CASE A
Let us assume that a textile producer is manufacturing 30 types of cloth
(different weights, weaves, color, etc.). Following the procedure for Case
A:
Step 1 : The 30 items are entered in Table I, with the values of (/ — O) ,
P, and S. (Also a rank number based on S.)
Step 2: IPS is calculated for each item and entered in Table I.
Step 3: Rearrange the items by sales rank number as shown in Table II.
Step 4: Using a subgroup of 2 for this case, compute X, R, and the
control limits, as shown in Table II.
Step 5: Plot the chart (Fig. 2).
Tt\o>^vo^'Oco(MnincN\oN«Doin(sn'OnN
<N — — < Nr _ r __ CN CN CNOO (N CN — CN ■— CM
CO O
Upper Control Limit
Average
Lower Control Limit
Items in Order of Increasing Sales Rate
10 20 30
Figure 2. Example of Control Chart, Case A
When more than 60 items are involved, it may be desirable to increase
the subgroup size to 3, 4, . . . 10. In accordance with control chart theory,
subgroup size should not exceed about ten when ranges are used, and a
subgroup size of two to five is preferable in this application.
INVENTORY POLICY BY CONTROL CHART 299
Several conclusions may be drawn from the IPS control chart by the
plant manager:
a) Since the average value of IPS in this example is 2.131, the sum of
monthly production and inventory is approximately twice the average
sales rate over the 30 items.
b) Since the lower control limit on the chart is less than unity, there
is a possible risk of certain items running out of stock, with inventory
plus production running behind sales, although no specific item had
an IPS less than 1.00 on the chart.
c) Item 4 has an IPS value above the upper control limit, with items
19, 16, and 24 high enough to warrant an extra check on sales fore
casts and production planning, so as to bring IPS values closer to the
target average value desired by the manager on his next control chart.
It should again be emphasized that this control chart procedure is in
tended for use on individual values of IPS only. The use of small subgroups
is in accordance with usual control chart practice to estimate the expected
variation occurring among items having a relatively close current dollar
sales rate. If certain items are not turning over rapidly, the subgroups for
these items will then have internal variations of IPS value which will then
be compared on the chart with other subgroups of items which are turning
over rapidly. The plant manager may then find that one of two situations
occurs on his chart:
a) Various values of IPS may fall outside of control limits in a random
manner across the chart. Such a situation simply indicates that the
production rates for these items need to be corrected if they are to
be maintained in a condition of inventory control on the same policy
as the other items.
b) In another situation, the manager may find that all of the items at
one end of the chart are out of control with respect to those at the
other end. The manager must then decide whether he wishes to
maintain a consistent policy over the entire group of items, or
whether he actually has a single policy or whether he should select
the righthand end of his chart) which should actually have been
charted under a separate policy. This is the situation where the chart
tends to be selfcorrecting, so that it will tell the plant manager
whether he actually has a single policy or whether he should select
a breakpoint and change his inventory policy for a certain portion
of the products having an exceptionally high (or low) turnover rate.
300 INVENTORY CONTROL
It may be noted that the chart also calls the manager's attention to the
ranking of items in order of sales value, and obviously this relative ranking
will change from month to month. It appears to the writer that this type
of ranking for setting up the chart is the one fundamentally of greatest
interest and utility to the plant manager. The subgroup size selected for
setting up the chart is only for the purpose of computing upper and lower
control limits for individual values, since the averages of subgroups are
not plotted. In case a plant manager who is statistically minded, should
question whether this method of grouping is in accordance with normal
control chart theory, we suggest that the purpose of the chart itself is to
tell him whether the values of IPS, so arranged, form an approximately
normal universe of values. If they do not, then considerable numbers of
IPS values will fall outside of the control limits, which indicates that the
manager does not really have a consistently successful inventory policy,
but only thinks he has, for the group of items being considered.
When the sales rate (S) of an item approaches zero, obviously the value
of IPS can have relatively large positive or negative values. This property
of IPS, together with the random fluctuations of sales, production dif
ficulties and delivery changes, makes the IPS factor a suitable and effective
parameter for impartial studies of inventory policy by the control chart
method.
Those who are familiar with the many other applications of control
charts will recognize that the values of E 2 given in this paper are based
on estimated limits extending three standard deviations above and below
the average. These limits are conservative and will only rarely give a false
indication that a value of IPS is too far from the target or central value.
If tighter limits are desired, the values of E 2 may be multiplied by % to
cut the limit to two standard deviations. When such narrower limits are
used, about 5 percent of the IPS values will tend to fall outside of limits
by pure chance, causing the plant manager a little additional worry. When
one is learning to use the charts, however, the narrower limits may promote
interest and result in a more rapid improvement in overall inventory control.
PART III
Facilities Planning
Chapter 1 5
INTRODUCTION TO PART III
Although often not the most frequent of operating decisions, facilities
plans can be the most significant. The facilities plan affects many other
decisions, not the least consequential of which are the production and
work force employment plans. The facilities plan establishes cost structures
and capacity constraints and, in so doing, goes a long way toward deter
mining optimum production plans.
The relationship between cost structure and capacity constraints on one
hand and production plans on the other was examined in Part I. It is
sufficient to suggest at this time that production plans are usually de
veloped with facilities planning in retrospect; that is, the facilities and their
resultant cost structures and capacity constraints are considered known
values. This sequence represents the normal procedure, since in most cases
the facilities are already in existence when production plans are laid.
However, in Part Three we will consider the entire facility a variable and
will examine the process of evaluating alternative facilities to create the
most favorable cost structures for future production plans. Moreover,
the relationship of work force employment plans and given cost structures
was examined through linear decision rules in Chapter 5. In this Part,
the facilities will be considered as they influence an ideal cost structure
which in turn determines work force employment plans through the deriva
tion of linear decision rules. In short, Part I used facilities as known
values and determined work force and production plans accordingly;
whereas in this part neither the facilities nor the production plans are con
sidered fixed.
Facilities planning is not restricted in this analysis to plant location
and plant expansion decisions; rather, facilities planning includes any
decisions which affect the makeup of the productive plant. Each equipment
305
306 FACILITIES PLANNING
addition or replacement decision constitutes facilities planning because
these decisions in their aggregate recast the overall production facility
and modify the cost structures and capacity constraints used in formulating
production and employment plans. The individual facilities decisions and
deliberations can, over an extended period of time, modify the facility sub
stantially and, therefore, are just as much facilities planning as are new
plant and relocation decisions.
Too often individualized equipment decisions are not considered facilities
planning and are regarded as separate engineering or modernization proj
ects. It is our purpose to place these decisions in a general management
planning framework.
Conventionally these decisions are determined in a capital budgeting
framework. That is, the facilities alternatives are considered as financial
investments and evaluated accordingly. This approach emphasizes the
profitability of alternative facilities plans under the assumption that the
alternatives which exhibit the greatest profitability will be undertaken.
In this way, it is thought, only the facilities plans which are superior in
an investment sense are adopted. Moreover, only the facilities plans which
are more profitable than financial investments would be considered. Im
plementation of this approach tends to budget the available capital to
highreturn investments and should therefore result in the creation of
the optimum type of facility for production and employment planning, be
cause only the facilities plans which show a profitable return will be under
taken. In other words, facilities plans are screened on the basis of prof
itability and only the most profitable alternatives are accepted.
This approach places great importance on the measure of profitability,
because profitability determines whether alternatives are accepted or re
jected. Any projects which are accepted and built into the ongoing pro
duction facility modify the existing cost structure and capacity constraints
and, therefore, constitute facilities planning according to our definition.
The measure of profitability used in the screening process is, therefore, of
crucial significance because it can determine which facilities plans are
undertaken.
The simplest and most common measure of profitability used in the
capital budgeting approach to facilities planning is the simple "payback"
period. The payback period is the number of years (or, more generally,
the period of time) required for the proposal to recover through cost savings
the amount of capital invested. The proposed alternative or plan usually
is a lower cost alternative than the present facility used in performing the
same function and generally requires an added investment. The relative
magnitude of the added investment compared to the generated cost savings
is a measure of the profitability of the alternative.
INTRODUCTION TO PART III 307
Specifically, if
/ = the added capital required by the proposed alternative;
N = the net cost savings generated by the alternative; and
P = the payback period measure of profitability; then
P =  (15.1)
N
Obviously the alternatives with a large / and a small N are relatively un
desirable; while those with a small / and a large N are more favorable.
Therefore, the smaller the payback period, P, the more favorable the plan.
If N is in dollars per year, P is also in years.
P has the advantage of being simple to calculate and compare with
alternative facilities. Its meaning is apparent. For example, if P = 1, the
alternative will "pay for itself," that is, recover the /, in one year. If, how
ever, P = 5, five years will be required to recover the /. P has the dis
advantage of being entirely relative. To verify this statement, ask yourself
what P is good and what is bad. P = 3 is good in some industries and bad
in others. The auto industry which completely retools every two years
might not be interested in a facilities plan wherein P = 3 ; however, in the
petroleum industry, a project with P = 3 might be considered very favor
able. In addition, P = 3 has no significance in comparison with financial
investments wherein the profitability is measured in terms of rate of return.
P does not, therefore, provide an adequate criterion to budget capital be
tween facilities plans on one hand and financial plans on the other. In
summary, the payback measure is simple and straightforward, but is not
sufficiently powerful to handle the general facilities planning — capital
budgeting problem.
The second measure used in conventional approaches, the simple rate
of return, makes up for some of the disadvantages of the P measure. First
of all, the simple rate of return compares the net cost savings to the
average amount of added capital in terms of the percentage rate of return.
This percentage rate of return can be compared to alternative financial in
vestments and thereby provides a criterion which is more generally ap
plicable than P. If:
R = the simple rate of return as a percentage per year,
R = ^j^ (15.2)
In this measure, the net earnings N are reduced by a factor A which
specifically allows for the recovery of the added investment /. If N and
308 FACILITIES PLANNING
A are constants per year, the average investment is — . Therefore, R is
a percentage rate of return which measures the profitability over and above
recovery of the added investment /. R can, therefore, be compared directly
to the rate of return of interest bearing securities or other financial in
vestments. A is usually determined by dividing / by the shortest life of the
plant so that R is the rate of return under the worst conditions. If the
actual life exceeds that used in computing A, the rate of return R would
be higher in actuality than originally computed. Specifically,
A — L (15.3)
T
when T equals the shortest possible use life.
Unfortunately neither of these criteria explicitly considers the time value
of money. Under the assumption that any cost savings from the first year
will be reinvested at the current rate of interest, $1 now is worth more
than $1 a year from now. In fact, $1 now is worth $1 f $1/ a year from
now when i is the prevailing rate of interest. Clearly the original dollar
has grown to include interest earned. Two years from now, the original $1
will be worth $1 + $l/ + /($i +$1/) =$1 +2i($l) + * 2 = (1 +0 2 . 1
Turning this concept around, a flow of earnings (or cost savings) of $1
per year for each of five years is worth less than $5 now, the product of
5 times $1, because some amount / (less than $5) invested now at i per
cent rate of return will yield a flow of $1 per year for five years. For
example, $3,791 invested now at 10% will yield $1 per year for each
of five years. The formula used in determining this value is:
Vz= y  = $3,791
t=i (1 + .10)*
In general terms, the present value, V , of N dollars per year for t years is:
V= £ N (15.4)
Formula (15.4) is used to determine elaborate tables found in most basic
financial management or managerial accounting texts to give V for various
values of t and i. For example, to select a few applicable values, one
such table contains V equivalents for flows of earnings reinvested at 1
percent to 50 percent for one to fifty years. Such tables are of immeasurable
value in avoiding the computation of V in accordance with formula (15.4).
1 This method of computation assumes that the $1 is received and reinvested in
the first day of the current year.
INTRODUCTION TO PART III
309
TABLE I
TnvesteH now
at % is equal to
—
per year
for
years
$3,791
10
$1
5
$6,145
10
$1
10
$7,606
10
$1
15
$8,514
10
$1
20
$9,915
10
$1
50
$2,991
20
$1
5
$4,192
20
$1
10
$4,675
20
$1
15
$4,870
20
$1
20
$4,999
20
$1
50
$2,035
40
$1
5
$2,414
40
$1
10
$2,484
40
$1
15
$2,497
40
$1
20
$2,500
40
$1
50
Source: Robert N.
1956, p. 496.
Anthony, Management Accounting, Richard D. Irwin, Inc.
Certain relationships are apparent from Table I. Given i, the present
value of $1 per year increases as N increases, but not in proportion to N.
For reasons previously examined, V increases less rapidly than N because
of the added value of reinvested earnings. Correspondingly, given N, V
decreases as i increases, again because of the reinvestment effect. A de
tailed examination of this presentvalue concept is outside the scope of
this book, although it is certainly a topic which is usually examined in
basic accounting, managerial economics, or financial management texts.
The concept of present value analysis is applied to facilities planning
problems by employing an economic model (or mathematical expression)
of the planned facilities investment. In this approach, a more explicit
formulation of the plan is used to specify the criteria of the decision more
carefully.
The value of any facility is equal to the earnings stream it generates
plus its market or disposal value, less the amount of added investment
required to produce the earnings stream. If:
V = economic value
D = disposal value
/ = investment cost
N = earnings stream
T = the economic or use life, whichever is less,
then
V = NT + D — I
(15.5)
310 FACILITIES PLANNING
But (15.5) is not in present value terms; that is, the flow and disposal
value are in terms of their value when they accrue rather than as of the
present time. To convert (15.5) to present value, we need only recall
(15.4) which expresses the present value of a flow of cost savings NasF =
T ft
2 •• Similarly, the present value of the disposable market
t=i (1 +/)*
value is V = when D t is the disposable market value as of the
(1 + 0*
point in time t, for example, as of the end of the fifth year. Finally, since
/ is already in present value terms,
t N D t
V= S 1 / (15.6)
t~i (1 + 0* (1 + 0*
Expression (15.6) gives the present value of a facilities investment assum
ing that the funds are reinvested at the beginning of each year t to earn
the rate of return i. To employ (15.6), we need only determine the sum
of the present value less the investment cost. If this value, V, is positive,
using an i equal to the cost of capital, then the facilities investment should
be favorably considered. When several alternatives are considered, the one
with the largest V should be accepted.
In order to make the V of model (15.6) more precise, we can consider
the effect of reinvestment (or rediscounting) instantaneously rather than
at the beginning of each year under analysis. To accomplish this precision,
we can introduce the relationship that
— > f Ne U dt + De~ iT
 : 1 +
which indicates that the calculus integral expression is approximated by
the formula we have been using. The integral expression rediscounts in
stantaneously the flow of earnings N at the rate of interest i over the time
period T. Expression (15.6) becomes
T
V— Ne U dt + De~ iT  / (15.7) 2
o
which is an exact economic model for a facilities investment.
2 Based upon an analysis in Chapter 12 of Bowman and Fetter's, Analysis for
Production Management, Richard D. Irwin, Inc., revised 1961.
INTRODUCTION TO PART III 311
To maximize V, we set the derivative of (15.7) equal to zero and solve
for T. 3
dV
— — Ne~ iT + D f e~ iT + De~ iT ' — i — =
dT ^ ^
Therefore, V is maximized when
Ne~ iT + D'e iT = iD T e~ iT
Dividing through by e~ iT ,
N + D' = iD T (15.8)
Expression (15.8) says that V is a maximum when the net earnings less
the change in disposable value (D' is decreasing) equals the interest on
the disposable value. In this expression, iD is the opportunity cost of re
taining the investment or the income which could have been obtained by
not making the investment. Therefore, as long as the net earnings less the
decline in disposable value exceed the opportunity cost (iD), more can
be obtained by undertaking the facility plan than by rejecting it. Expres
sion (15.8) would be used to determine T given N, D t and i, and that
value would be used to determine the V in ( 15.7) . If the V thus determined
is positive, the investment is acceptable.
It would be useful to reconcile (15.7) and (15.2), since the simple rate
of return (15.2) is a special case of the present value analysis (15.7). By
definition, the simple rate of return does not consider the present value
of money and considers Af constant over time.
Therefore,
V— 2 N + D — l = TN + D T — I
t=i
Furthermore, in the simple rate of return analysis, the rate of change of
salvage value is — (assuming straight line amortization of /). Therefore,
D T = I—TD = I— T— =0
T
*=**! Therefore, ± r« = r» — (*T) = e^
dx dx dT dT
312 FACILITIES PLANNING
Finally, the rate of return over time period T equals J[ where — is the
average investment. But
Therefore,
V = TN + D T — I
= TN +  /
= TN — I
R _J^ ZZ TN — I
I I (15.9)
Dividing (15.9) by J,
/
# = (15.10)
Y
Thus the R of (15.10) is expressed as a percentage per year; further
more the — of (15.10) equals the A of (15.2) and the conditions of
(15.10) and (15.2) are identical. In other words, the conditions which
apply to the simple rate of return analysis (15.2) apply if (1) the
time value of money is ignored, (2) the rate of amortization, A, is con
stant and equal to — , and (3) the disposable value, D f , is zero at time T.
In order for the disposable value to be zero at the end of period T
(the economic life), the use life must be at least equal to T. The use life
is equal to the period of time from installation to obsolescence or complete
failure, whichever is less; and the economic life is defined as the period
of time when V is maximal. Therefore, in order for the simple rate of
return to apply, the facility investment must reach a maximum value and
be declining when failure or obsolescence sets in. Under these conditions,
plus conditions (1) and (2) listed above, the simple rate of return can
be used as an approximation of the present value model.
So much for the conventional treatment. It is apparent that none of
these traditional models considers certain conditions which are present in
many operating decisions in the facilities planning area. The first of these
is the problem of uncertainty. The net earnings, N, are a significant variable
in determining the present value, V, of the facilities investment and, hence,
INTRODUCTION TO PART III 313
the simple rate of return. However, N can only be estimated in any plan
ningtype analysis. And yet there is always some uncertainty in forecasting
a value for N. In other words, perfect forecasting is impossible; hence,
uncertainty obtains in the facilities plan. Any general formulation of the
facilities planning problem must incorporate a means of adjusting to various
forecasts. Chapter 16 treats this problem by putting facilities planning in
a game theory (uncertainty) format, an approach which is further general
ized by an editorial note.
The second gap in the conventional approach concerns analyzing the
overall effect of individual decisions. Each factory is part of a general
system which processes materials and distributes the resulting products.
Individual facilities plans must be built into the system of plants which
comprise the entire firm's production facility. Chapter 17 treats this
problem by developing a linear programming model for the interrelated
units aspect of facilities planning.
Finally, none of the conventional measures considers the general
problem of determining the total size of a facility. The application of capital
budgeting methods will result in planning the most efficient facility given
a level of demand, because the net earnings, N, are dependent upon a
particular demand forecast. The conventional procedure will not neces
sarily accept plans to expand facilities in anticipation of forecasted increases
in output. This problem area is called scale (or size) of operations and
is analyzed in a fairly general form in Chapter 17.
These three chapters, therefore, constitute a reasonably complete analysis
of the general facilities planning problem.
Chapter 1 6
UNCERTAINTY PROBLEMS
IN FACILITIES PLANNING
XVIII.
Capital Budgeting and Game Theory*
Edward G. Bennion
The purpose of this article is to do two things: (1) to suggest a more
rational approach to capital budgeting, which is a perennial and imper
fectly solved problem for business, and (2) to test the applicability of
game theory to the kind of decisions which are involved in capital budgeting.
In a sense, this is singling out a particular problem and a particular
technique. However, the problem happens to be one of the most im
portant and least clarified of the toplevel issues faced by businessmen, just
as the technique happens to be one of the most intriguing and least
understood of the statistical devices which have recently been presented to
businessmen as aids to toplevel decision making. In combination, they
offer an unusual opportunity to push progress ahead in an area where it is
needed and at the same time put some realism into a methodology whose
value may be overestimated at its present stage of development.
ROLE OF FORECASTS
The whole subject of capital budgeting is, of course, too big and com
plicated to be critically examined within the scope of one brief article. But
*From the Harvard Business Review, Vol. 34, No. 6 (1956), 115123. Reprinted
by permission of the Harvard Business Review.
314
CAPITAL BUDGETING AND GAME THEORY 315
perhaps some new light can be thrown on one important problem aspect:
the relationship that should obtain between the budget and the economic
forecast.
It hardly seems necessary to prove that economic forecasts play a
significant — sometimes an almost determining — role in shaping the business
man's investment decisions. The traditional explanation is simple:
In order to decide how much a company should invest or what kinds of
assets it should acquire, we need a sales forecast for the firm — to establish its
anticipated level of activity.
But the firm's sales forecast cannot be made without some estimate of what
the industry is going to do. And the industry's sales forcast in turn depends in
large measure on the predicted level of activity in the economy as a whole.
Q.E.D. — the capital budget of any individual firm has a unique and important
relation to the general economic forecast.
UNRELIABLE GUIDE
If it is obvious that forecasts are necessary, it is still more obvious that
they are likely to be unreliable:
It is impossible to make an economic forecast in which full confidence can
be placed. No matter what refinements of techniques are employed, there still
remain at least some exogenous variables — i.e., variables, such as defense ex
penditures, the error limits of whose predicted values cannot be scientifically
measured.
It is thus not even possible to say with certainty how likely our forecast is
to be right. We may be brash enough to label a forecast as "most probable,"
but this implies an ability on our part to pin an approximate probability co
efficient on a forecast: 1.0 if it is a virtual certainty, 0.0 if it is next to an
impossibility, or some other coefficient between these extremes. But, again,
since we have no precise way of measuring the probability of our exogenous
variables behaving as we assume them to do, there is no assurance that the
estimated probability coefficient for our forecast is anything like 100% correct.
In spite of such drawbacks, businessmen are willing to pay for having
general economic forecasts made, and to use them in deciding among
alternative investment opportunities for capital funds. For example, the
more certain is prosperity, the wiser it will usually be to invest in new
plant and equipment; whereas the more certain is depression or recession,
the safer it looks to invest in government bonds or other securities. In
other words, the businessman uses economic forecasts, to assess the rel
ative advantages of investing in fixed or liquid assets, in the light of the
expected businesscycle phase.
316 FACILITIES PLANNING
COMMON ERROR
At this point, the businessman stands before us, his economic forecast
in one hand, his proposed investment alternatives in the other. His next
step is the one where he is most apt to go wrong. When some one phase
of the business cycle is forecast as "most probable," it is likely to look
logical to him to go ahead and put his funds into whichever investment
alternative maximizes profit in the phase expected.
Looking at the situation superficially, this step appears to be quite
sensible. But actually a businessman armed with only a single most prob
able forecast is in no position to make a wise investment decision — unless
his forecast is 100% correct, and this, as we have seen, is an impossibility.
A NEW APPROACH
In the following pages, a more rational way to use an economic forecast
is suggested. Furthermore, adoption of the method proposed here permits
the businessman to learn the answer to another question over which he
probably has spent some sleepless nights if he has ever known responsi
bility for making a decision on the capital budget. Just how far off can
the forecast be before it leads to a "wrong" investment decision?
Because the fundamentals of this new approach are most easily grasped
if a specific problem is attacked, let us see how it can be applied in con
crete cases. We shall look first at a simplified hypothetical case, and then
at a case based on actual experience (slightly disguised). For the sake of
the clearest possible focus on the problems involved, no explicit reference
will be made to the role of game theory while we are working out their
solution. Following their presentation, however, we will meet the theory
headon and discover in the process that we have already drawn from it
just about as much as is possible.
SIMPLIFIED CASE
This first case, although hypothetical, is not unrealistic. Further, it has
the advantage of reducing the problem and method of solution to the
simplest possible proportions.
ALTERNATIVE INVESTMENTS
The specific issue of whether to invest in plant or securities is a good one
for illustrative purposes because it can be defined so sharply. Suppose we
have even more exact information than most businessmen generally as
semble before exercising their judgment to reach an investment decision.
CAPITAL BUDGETING AND GAME THEORY
317
Under these circumstances we might know that:
The most probable forecast is for a recession.
In recession, investment in plant will yield 1 % as compared with a 4%
yield for securities.
In prosperity, plant will yield 17%, while securities will yield 5%.
Placing these data in diagram form, we get the following 2x2 "matrix":
Management
Investment
Alternatives
Securities
Plant
CyclePhase A Iternatives
Recession Prosperity
4%
5%
1%
17%
Under this condition no businessman worth his salt is going to want
to settle for securities — but how can he justify any other course, given
his forecast of a probable recession?
MORE DATA NEEDED
To begin with, our businessman needs to recognize that the data so
far placed at his disposal, rather than limiting his choice, do not provide
the basis for a decision at all. Two further questions first require an
answer:
( 1 ) How probable is the "most probable" forecast? To answer this, the fore
cast needs to be complete by assigning a probability coefficient to each cycle
phase considered.
(2) How probable does a recession have to be before the earnings prospects
of the more conservative choice look just as attractive as the returns available
from adopting a bolder course of action? In other words, what are the in
difference probabilities of recession and prosperity, given the rate of return
each will yield?
PROBABILITY COEFFICIENTS
Establishing probability coefficients on the economic forecast is a job
we can relinquish, more than willingly, to the company economist. We are
not concerned here with what kind of crystal ball he gazes into, but rather
with how top management uses his findings, whatever they may be. So,
in order to get on with our problem, let us simply suppose that our fore
caster thinks the chances of recession are 6 out of 10 and so has assigned
a probability coefficient of 0.6 to his predictions for a recession (which
automatically means 0.4 for prosperity).
318
FACILITIES PLANNING
INDIFFERENCE PROBABILITIES
The handling of indifference probabilities is not going to be quite so
simple, but it can be done readily enough by anyone who can recall his
high school course in algebra (he does not have to be blessed with
"total recall," either) :
Suppose we say, in elementary algebraic terms, that the recession probability
coefficient = R, and the prosperity probability coefficient = P. In that event, we
know from our matrix figures that over any period :
I: 4R + 5P = the return on securities
II: IR + UP = the return on plant
From this, it is clear that the return on securities will be the same as the
return on plant when:
III: 4R + 5P= 1R+ UP
Solving this last equation for R in terms of P, we get:
IV: R = 4P
Since the sum of the probability coefficients (R + P) has to equal 1.0, we
can say that R = (1 —P) and substitute (1  P) forR in IV:
V: 1 P = 4P, or
VI: P = 0.2, and R = 0.8
This merely means that if the probabilities of a recession and prosperity
are 0.8 and 0.2 respectively, then the chances are that the company will
be just as well off investing in securities as in plant, and vice versa. In
other words, it appears to be a matter of indifference which alternative is
chosen.
ASSEMBLING THE DATA
For the sake of convenience, let us reassemble all our information in
the compact easytoread form of a matrix:
Management
Investment
A Iternatives
Securities
Plant
Indifference probabilities:
Forecasted probabilities:
CyclePhase A Iternatives
Recession Prosperity
4%
5%
1%
17%
/? = 0.8
R = 0.6
P = 0.2
P = 0.4
Here, at last, our businessman has all the information he needs to decide
CAPITAL BUDGETING AND GAME THEORY 319
what course of action maximizes his chances for success. So long as the
forecasted probability coefficient for a recession is not equal to or greater
than the indifference probability coefficient for this phase of the business
cycle, the businessman can know that he is not making an avoidable mis
take by playing for high stakes and building a plant. The alternative to
choose is the one that has a higher forecasted probability than indifference
probability.
ADVANTAGE GAINED
The little technique outlined above thus does two things:
1. It makes clear that the best paying investment alternative in the most
probable situation is not necessarily the alternative that management should
choose.
2. With indifference probabilities, it is possible for us to see what margin
of error is permissible in any estimated probabilities before these estimates
result in an erroneous decision.
ACTUAL CASE
With this much understanding of the 2 X 2 matrix, we are now in a
position to apply indifference probabilities to our actual but more com
plicated case:
An integrated petroleum company anticipates the need for a refinery in
Country A, has determined that Alpha City is the best location, but is un
certain as to the appropriate size.
Operating at approximate capacity, internal economies of scale exist up to a
refinery size of R barrels per day (B/D).
However, once sales exceed Z barrels per day (with Z < R), further econ
omies could best be effected by building a second refinery elsewhere. This
puts a ceiling of Z barrels per day on the Alpha City unit. 1
Whatever size refinery is built, it can be completed in 1965. It is also agreed
that depreciation and obsolescence will make the refinery valueless by 1978.
As an aid in determining the size required in 1965 , it is known that con
sumption growth is highly correlated with industrial output.
Unfortunately, there is less than perfect unanimity as to the expected
growth rate of industrial output between the present and 1965. The economics
department has forecasted a rate of 2.5%, the foreign government officially
estimates a rate of 5%, and the company top management wonders what
would happen if the growth rate turned out to be 7.5%.
1 This ceiling decision, it might be noted, involves the solution to a problem to
which linear programing conceivably might aptly be applied. Taking this solution as
given obviously does not mean it is necessarily easy to come by. See Alexander
Henderson and Robert Schlaifer, "Mathematical Programming: Better Information for
Better Decision Making," reprinted in this book on page 53 .
320
FACILITIES PLANNING
Careful analysis leads to the conclusion that a growth rate of 2.5% requires
a refinery of X barrels per day capacity; that a growth rate of 5% necessitates
a refinery of Y barrels per day capacity; and that a growth rate of 7.5% re
quires a refinery of Z barrels per day capacity, this last being our previously
established ceiling size.
Pending further study, all agree to work on the assumption of a zero
growth rate after 1965.
To evaluate the three alternative refineries, anticipated integrated in
come (covering refining, marketing, producing, and transportation) will be
computed for each facility under each growth rate, and the percent return
on integrated investment will then be calculated and compared. 2
MATRIX AND PROBABILITIES
With three sizes of refineries to consider, and three growth rates, we
will get a 3 X 3 matrix on this problem. The figures on the diagram, repre
senting return on integrated investment, are more or less what common
sense tells us to expect. For example:
The small X BID refinery shows the highest rate of return if the growth
rate is a low 2.5%, bringing in an 8.8% return against only 2% for the
large Z BID unit with its much higher cost of investment.
On the other hand, if the growth rate should reach a high of 7.5%, the
large Z BID refinery can return an average of 12.6%, while the small X
BID facility with its limited output is tied to its 8.8% ceiling yield.
But filling in the matrix does more than verify our commonsense con
clusions. It also equips us to see how much better one refinery is than
another under each possible condition.
Pre1965 Growth Rate Alternatives
Low
2.5%
Moderate
5.0%
High
7.5%
ZB/D
Refinery
Investment Y B/D
A Iter natives
XB/D
Indifference
probabilities:
Forecasted
probabilities:
2.0%
7.3%
12.6%
3.7%
11.0%
11.0%
8.8%
8.8%
8.8%
0.301 M = 0.114 H = 0.585
L 0.333 M = 0.333 H = 0.333
2 While the matrix figures of this case are based on careful engineering estimates,
this venture is still in an experimental stage and hence does not constitute a part of
the budget procedure of Standard Oil Company (New Jersey), with which I am
associated.
CAPITAL BUDGETING AND GAME THEORY 321
The indifference probabilities here were calculated by just the same
algebraic procedures as were followed in our previous example. The fore
casted probabilities simply reflect the fact that no one in the com
pany could decide which of the three forecasts was most likely, and there
fore each was treated as equally "valid" (i.e., chances of 1 out of
3, or O.331/3).
THE SOLUTION
A quick look at our diagram now reveals that the extralarge Z B/D
refinery should be ruled out, since its return will be greater only under
a 7.5% growth rate, and the real probability for a 7.5% growth rate is
too small to justify considering that alternative. (Remember that an alter
native cannot be chosen unless its forecasted or estimated true probability
is equal to or above the indifference probability.)
On the other hand, both the X B/D and Y B/D refineries are still in
the running. So, with two possibilities still remaining, new indifference prob
ability calculations are needed in order that we may choose between them:
Suppose (as before) we use the letter L to represent the indifference prob
ability for the low 2.5% growth rate; M for the moderate 5% rate; and H
for the high 7.5% figure. In this event we read off the matrix that:
I: 3.7L+ 11.0M+ 1 1 .OH = the return on the Y BID refinery
II: 8.8L + 8.8 M + 8.8// = the return on the X BID refinery
From this it is clear that the return on Y BID will be the same as the re
turn on X BID when:
III: 3.7L+ 11.0M+ 11.0// = 8.8L + 8.8M +8.8//
Solving this equation in terms of L we get:
IV: L = 0.43 (M + H)
This means that if the estimated true probability of a 2.5% growth
rate is greater than 0.43 of the combined estimated true probabilities of
the 5% and the 7.5% growth rates, the X B/D refinery is a better bet
than the Y B/D refinery. This has to be true because the X B/D refinery
is the bestpaying alternative, given the 2.5% growth rate.
Since the estimated true probability for the 2.5% growth rate is actually
0.33, which is slightly greater than 0.43 X (0.33 + 0.33), we would
conclude that the X B/D refinery is a little better bet than the Y B/D
refinery, with the Z B/D refinery showing a very poor third.
A CHANGED ASSUMPTION
Now that we have reached an answer to the problem as originally stated,
322 FACILITIES PLANNING
let us (realistically if provokingly) proceed to alter some of our assump
tions, and see just what this will do to our choice:
Suppose that top management, having injected the 7.5% growth rate into the
original problem for comparative purposes, concludes that the probability of
a 7.5% growth rate is really nil, and that the probabilities of the 2.5% and 5%
growth rates are each 0.5. The indifference equation then becomes:
3.7L+ 11.0M = 8.8L + 8.8M, or L = 0.43M
Since the estimated true probability of 0.50 for 2.5% growth rate is,
in this instance, a great deal bigger than 0.43 X 0.50, we would conclude that
the X BID refinery is a lot better bet than the Y BID refinery, and that no one in
his right mind would even consider building a Z BID unit.
Thus we come to the same general conclusion as before, only a bit more
cocky, as a result of writing off the 7.5% growth rate and distributing its
former probability in such a way as to make the 2.5% and 5% growth
rates equally probable.
A RADICAL REVISION
It is clear, however, that the above conclusion is suspect unless we
expect no economic growth in Country A after 1965. If we do expect
further growth, the X B/D refinery loses much of its $sign allure. It will
not be large enough to take advantage of Country A's expanding economy
and so can never return any more than 8.8% on investment.
In contrast, the Z B/D refinery will show an increasing rate of return
as A's expanding market permits it to produce more and more per year,
perhaps ultimately reaching its capacity. Thus, instead of spurning the Z
B/D refinery (as in our last example), we must acknowledge its potential
attractiveness — provided A's economy does not get stalled after 1965, as
was previously assumed.
With this possibility in mind, let us make some alterations in our problem
and see what we should do. There are two new conditions:
1. After 1965, it is now agreed, the growth rate of Country A will be a
steady 2.5% each year.
2. It would be possible to build an X BID or Y BID refinery that would
be expansible to Z BID; such units would cost more than nonexpansible
facilities, but less than two separate refineries with a combined Z BID capacity.
At this point our real problem becomes one of deciding whether to
build a Z B/D refinery or an X B/D ora 7 B/D refinery expansible to
Z B/D. It may be helpful, first, to consider how the figures in this new
matrix should differ from those presented earlier:
Most of the figures are higher than before, reflecting the fact that average
CAPITAL BUDGETING AND GAME THEORY
323
earnings are increased by higher sales toward the end of the productive life
of each unit.
To a limited extent, the higher investment costs of the two expansible units
operate as a drag on their earnings. Thus two of the figures happen to be
lower than before, and the expansible refineries have a lower maximum re
turn than is possible with a Z BID unit.
On the whole, the figures in the matrix tend to be squeezed closer together;
i.e., they no longer range between such wide extremes.
Now our revised matrix reads like this:
Refinery
Investment
Alternatives
Indifference
probabilities:
Forecasted
probabilities:
ZB/D
Expansible
YB/D
Expansible
XB/D
Pre1965 Growth Rate Alternatives
Low Moderate High
2.5% 5.0% 7.5%
4.6%
9.9%
12.6%
7.2%
10.5%
11.1%
8.0%
9.7%
10.4%
L 0.338 M 0.129 # = 0.533
L 0.333 M 0.333 # = 0.333
Again, we work our algebraic equations to find the indifference prob
abilities, while the forecasted probabilities result, as before, from assigning
equal weight to each forecast.
In this instance, the expansible Y B/D refinery is a shooin. This can
be intuitively seen by recognizing that the Y B/D refinery is the best
paying one given the 5% growth rate, and this growth rate is the only
one with an indifference probability lower than its estimated true prob
ability. By somewhat similar reasoning, the Z B/D refinery is distinctly
the worst choice.
The new matrix also reveals another significant conclusion. One does
not have to be a mathematician to perceive, just from inspection, that the
cost of a poor decision here is a good deal lower than it was for our
earlier versions of this problem. The result, of course, flows from the
fact that the differences in the row and column vectors have been greatly
narrowed. The absolute cost of a mistake is by no means insignificant,
324 FACILITIES PLANNING
but relatively it is much less than in the previous matrix. This piece of
information is in itself of considerable value. At a minimum, it will help
the budgetmaker to do less agonized tossing in his bed.
MORE TINKERING
Just for fun, let us tinker with our problem once more before dropping
it, and again assume that management rejects as of nil probability the
growth rate of 7.5% between the present and 1965, giving the 2.5%
and 5% growth rates equal probabilities (i.e., chances of 1 out of 2, or
0.5):
This eliminates the top row and right column of the 3x3 matrix, leaving
it 2 X 2.
The indifference probabilities equation for the X BID and Y BID refineries
then becomes:
8.0L + 9.7M = 7.2L + 1 0.5M, or L = M
This means that the indifference probabilities for both L and M are 0.5.
These, however, are also the values for the estimated real probabilities for
the 2.5% and 5% growth rates.
Consequently, we have here the unusual case in which the X BID and Y BID
refineries are equally good bets, with the Z BID refinery being no bet at all,
No matter which way we look at the problem, therefore, the Z B/D
refinery is the poorest choice. But, under one probability assessment, the
expansible Y B/D refinery is a better choice than the expansible X B/D
refinery; under the other, the expansible X and Y B/D refineries are toss
ups. This conclusion is, of course, decidedly different from that reached
for the previous matrix, in which the influence of post 1965 growth was
ignored.
OTHER BUDGET QUESTIONS
So far we have managed to explore only one small corner of the
capital budget domain. Let us look at some further problem areas.
PROBLEM OF TIMING
Our new technique can also be put to work on a timing problem. Since
all but one of the figures in the following matrix have been chosen some
what arbitrarily (although the choices can easily be defended), they call
for no explanation. The only exception is the 9.6% prosperity return for the
Y B/D refinery which appears in the lower righthand corner; this is the
weighted average return from this investment under the three possible
rates of growth, assuming equal probability for each.
CAPITAL BUDGETING AND GAME THEORY
325
Management
Investment
Alternatives
Indifference
probabilities:
Government
Bonds
Other
Securities
YB/D
Refinery
Pre 1965 CyclePhase Alternatives
Depression Recession Prosperity
3.5%
3.0%
2.5%
2.0%
5.0%
3.0%
2.0%
3.0%
9.6%
0.43 R = 0.24
0.33
It would be a tedious repetition of now familiar principles to attempt to
bleed this matrix dry. Let us content ourselves, therefore, with just one
reasonable (and relatively simple) interpretation:
Since depression is comparable to the 19371938 decline in this country,
we might well reject this as being of nil probability in the period between
now and 1965.
Should we do so, the left column of the 3x3 matrix would be eliminated,
as would the top row, since government bonds would not be a logical invest
ment except under depressed conditions.
In the remaining 2x2 matrix, the indifference equation for other securities
and our Y BID refinery then becomes:
5. OR + 3.0P = 3.0R + 9.6P, or P  0.3/?
In other words, unless we think the true probability of a recession is
something more than twice as great as that of prosperity, the construction
of the Y B/D refinery ought not to be deferred.
INVESTMENT PRIORITIES
Whenever more than one investment alternative is available, there arises
the problem of assigning an order of priority among them. To assess and
compare each possible project, the method followed in the previous prob
lem can be used to advantage again:
(1) Using a 2 X 2 matrix, calculate indifference probabilities for in
vesting in securities and in plant (or other assets to be used in the
company's own business).
(2) Repeat this process for each contemplated internal use of com
pany funds. The top row will be the same in all of these matrices, but the
bottom rows will not in general be the same. Consequently, the indifference
probabilities for the numerous matrices may vary widely. Any company
326 FACILITIES PLANNING
project with an indifference probability coefficient for prosperity that is
lower than the estimated true prosperity probability coefficient is a good bet.
(3) Instead of arraying the projects in order of descending return for
the most probable cycle phase (which would incur all the defects already
shown to exist in tying investment decisions to a single most probable
forecast), array them in order of descending weighted average return —
weighted according to the estimated probabilities of recession and prosperity
or of different rates of growth.
In general, this procedure will not result in the same priority order for
projects as the method commonly employed, but it is a better method
of evaluating all the alternative uses for funds. This is because it avoids the
frequently fatal mistake of betting on whatever venture seems to look
most profitable, given only a single most probable forecast.
If desirable projects turn out to be more numerous than company re
sources can finance, the management must then decide whether it wants
to borrow or not. If external financing should be ruled out, the marginal
project must be the one with the lowest weighted average return which
just exhausts available funds. On the other hand, if all desirable projects
do not exhaust available funds, the marginal project is the one whose
indifference coefficient for prosperity is just equal to the estimated true
prosperity coefficient. The excess funds should be temporarily invested
in securities.
ALMOST A GAME
Now, finally, we are ready to have that initial promise redeemed — i.e.,
that the role of game theory would be explained and be evaluated. Actually,
as anyone who has met this theory before will recognize, it has already
been introduced! All our matrices have been "games," although, in playing
some of them, we have had to construe a few of the rules pretty loosely.
The two players in most of our games have been the businessman and the
business cycle. Each has had either two or three "strategies." For the
former there have been different types of investment alternatives; for the
latter there have been different cycle phases. Indeed, the indifference prob
abilities calculated by the businessman for depression, recession, and pros
perity have an exact parallel in game theory. Those probabilities constitute
what would be known as the "business cycle probabilities" — namely, the
percentage of the time that the business cycle should provide each of its
phases in a random manner to hold the businessman's gains down to a
minimum.
Can it be said, then, that our method for deciding on the capital budget
marks an extension of game theory concepts to the field of business and
CAPITAL BUDGETING AND GAME THEORY 327
economics? In the strictest sense, the answer must be no. Ours is not a
rigorous game — it does not meet all the conditions requisite for such a
game:
In game theory proper, the opposing players are assumed to be completely
selfish and intelligent. Charity and stupidity are unknown to either. Clearly
the business cycle, however malevolent it may sometimes seem, does not meet
these requirements. It is as impersonal as nature. In fact, what we really are
doing in problems like ours is playing games with nature. Thus the indifference
probabilities in our last matrix are actually nature's probabilities. They tell us
that, if nature were malevolent, it could minimize its "losses" to the business
man by providing depression, recession, and prosperity, in a random manner,
43%, 24%, and 33% of the time, respectively.
If the businessman were confronting an opponent who could maximize gains
and minimize losses by a deliberate choice of strategies, then there would be
additional calculations to make and prohibitions to observe. For example, in
order to keep a selfish and intelligent antagonist from guessing what he might
do and benefiting by the knowledge, the businessman might have to figure
out several strategies for himself and then use them randomly. Thus, again in
our last matrix, the businessman's odds are such that he should invest in
government bonds, other securities, and the Y BID refinery, in a random man
ner, 90%, 2%, and 8% of the time, respectively. Otherwise, faced with a
malevolent nature, he would fail to maximize his gains.
It takes some stretching to make a choice of strategies out of a range of
possibilities, yet a range of possibilities is all we can get out of nature (as con
trasted with a willful opponent); and in the case of some business problems we
cannot even get that. Moreover, nature, alias the business cycle, may have some
strategies on the matrix that no sensible antagonist would use at all because
under all conditions other strategies would give him higher gains or lower
payoffs.
Consequently, our games with nature are not of the "purer," more
rigorous type. Our version represents a departure by virtue of recogniz
ing four additional facts: (1) nature is not malevolent; (2) the odds of a
malevolent nature are really the indifference probabilities of the business
man with respect to his alternative courses of action; (3) any time the
estimated odds of nature's strategies differ from the businessman's in
difference odds, there is a best strategy for the businessman; and (4) this
best strategy, as well as the degree of its "bestness," depends on the rela
tionship between the estimated and indifference odds.
However, any readers who are interested in further pursuing the rules
of the game theory proper can do so handily by consulting J. D. Williams,
The Compleat Strategyst (New York, McGrawHill Book Company, Inc.,
1954). Anyone who can add and subtract can follow this pleasant and
328 FACILITIES PLANNING
often humorous exposition, whereas most other books on the subject call
for more advanced mathematical learning. 3
OTHER ECONOMIC "GAMES"
If capital budgeting can only borrow from game theory but not take it
over in its entirety, what about any other business applications? It should
be possible to find in the businessman's competitive world a variety of
situations that resemble orthodox games — i.e., where the opponents are
not noted for their charity toward each other.
The existence of many such parallels is obvious, but unfortunately game
theory in its present state of development is not far enough advanced to
handle most of them. (Originated by von Neumann, the theory first
achieved a wide audience when he and Morgenstern published their book
in 1944. 4 ) Thus, game theory still does not deal effectively with situations
where there are more than two players or where the loser's losses and the
winner's gains do not cancel out. For example:
The most common realistic game cited in economic literature is a duop
olistic (twoseller) situation in which each of the duopolists has alternative
strategies and seeks the strategy that will maximize his profits. 5 This may
be a realistic example, but it is certainly one of limited existence. The
businessman may not have a large number of competitors but he usually
has at least several. However, to consider several competitors plunges us
into games involving more than two players, and here the theory as it
now stands leaves much to be desired.
Another possible realistic game on the twoperson level is where the
opponents are the businessmen and the trade union. But this sort of game
is likely to be one in which the solution may harm or benefit both players,
or harm one player more than it benefits the other. This throws us into
games with a nonzerosum payoff, where the theory again leaves much
to be desired.
To say that game theory, in its more rigorous sense, still has no sig
nificant business applications does not of course mean that claims for its
3 John von Neumann and Oscar Morgenstern, Theory of Games and Economic Be
havior (Princeton, Princeton University Press, 1944); J. C. C. McKinsey, Introduc
tion to Theory of Games (New York, McGrawHill Book Company, Inc., 1952);
David Blackwell and M. A. Girshick, Theory of Games and Statistical Decisions
(New York, John Wiley & Sons, Inc., 1954).
4 See footnote 3.
5 See L. Hurwicz, "The Theory of Economic Behavior," in George J. Stigler and
Kenneth E. Boulding (Editors), A.E.A. Readings in Price Theory (Chicago, Rich
ard D. Irwin, Inc., 1952), Vol. VI.
CAPITAL BUDGETING AND GAME THEORY 329
potential have been exaggerated. The day of orthodox game theory may
well be on its way, just as the day of linear programming has already arrived
in some measures. Meanwhile, businessmen may wish to acquaint them
selves with the theory and be on the watch for any practical uses it may have.
CONCLUSION
To summarize briefly, we have seen that forecasting can result in a
negative contribution to capital budget decisions unless it goes further than
merely providing a single most probable prediction. Without an estimated
probability coefficient for the forecast, plus knowledge of the payoffs for
the company's alternative investments and calculations of indifference prob
abilities, the best decision on the capital budget cannot be reached.
Even with these aids the best decision cannot be known for certain, but
the margin of error may be substantially reduced, and the businessman can
tell just how far off his forecast may be before it leads him to the wrong
decision. It is in assessing this margin of error, along with the necessarily
quantitative statement of alternative payoffs, that some of the concepts of
game theory make their particular contribution to the problem.
EDITORS' NOTE
Bennion presents his analysis using three facilities planning alternatives
and three economic forecasts. His framework is not generally applicable
because it employs specific alternatives and levels of economic activity.
We would improve his framework considerably by generalizing it as a
problem with m alternatives and n forecasts.
Let A u . . . ., A m describe various facilities plans, there being m such
proposals in general. Describe the forecast conditions as F u . . . ., F n where
there are n such forecasts. Finally, let p i} stand for the timeadjusted (dis
counted) rate of return for the i fh alternative paired with the j™ forecast.
Each p i} would be arrived at by computing the interest rate which equates
the earnings stream to the added investment. There would be m times n
such computations.
6 See Alexander Henderson and Robert Schlaifer, op. cit. (footnote 1).
330
FACILITIES PLANNING
In this general formulation, the capital budgeting problem is presented
as follows:
The problem is to select an alternative A t (where A lf . . . ., A m are mutually
exclusive). Among others, the following criteria can be used.
1. Select the A r for which the minimum p rj (j = 1, . . . ., n) is
greatest. That is, select the alternative for which the worst possible
return overall forecasts F{, . . : ., F n is maximized. This strategy
is called a MAXIMIN strategy because it will MAXimize the
MINimum return or MINIMAX because it will MINImize the
MAXimum disadvantage.
2. Select A r to maximize the maximum return over all forecasts
F u . . . ., F n . This strategy is an optimistic or MAXIM AX strategy
(which MAXImizes the MAXimum return).
3. If no objective probabilities exist for F 1 , . . . ., F n forecasts, as
sign an equal probability to each forecast and select the alternative
1.0
with the greatest payoff. In this case, p x — p 2 . ... p n
.The
payoff is ^PjPh This is an EQUIPROBABILITY strategy.
There are other game theory strategies which could be used to analyze
the uncertainty problem. 7 However, these three criteria illustrate how the
facilities planning problem can be formulated in uncertainty terms.
See Luce, R. D. and Raiffa, Games and Decisions, John Wiley & Sons, Inc., 1957.
Chapter 1 7
FACILITIES PLANNING WITH
MATHEMATICAL MODELS
XIX.
Mathematical Models in
Capital Budgeting*
James C. Hetrick
Some of the major responsibilities of top management are in the fields
of longrange planning and capital budgeting. Planning groups are con
stantly faced with such questions as:
Should we build a new plant?
If so, where is the best place to build it?
And when should it be built?
Or, instead of building, should we expand our existing facilities?
Should we also modernize them?
This broad field of decision making for capital investment is one of the
most difficult, one of the most recurrent, and one of the most controversial
of management areas. And it is also an area where there are tremendous
* From the Harvard Business Review, Vol. 39, No. 1, (1961), 4964. Reprinted by
permission of the Harvard Business Review.
331
332 FACILITIES PLANNING
opportunities for basic improvements in operations and policies. Here are
just a few of the shortcomings that show up again and again in corporate
practice :
Many companies have never asked themselves such important ques
tions as what the function of capital is in an industry.
Some managements pay only lip service to the idea that decisions
should be made to the best advantage of the total enterprise and for
the long term. All too frequently, shortterm decisions are made that
are crippling in the long term.
Capital is often allocated for the good of a department or for a cost
center rather than for the company as a whole.
Confusion is likely to result if executives are asked to define the
extent to which different investment decisions should be considered as
being independent of each other.
It is rarely recognized that the proper rate of return may be different
for various parts of the organization. In fact, many managements even
fail to discount for differences in useful economic life.
In recent years operations research has been getting much publicity in the
solution of the tactical problems associated with daytoday decision making
and immediate operations planning. However, the techniques of operations
research can also help management face the issues and arrive at decisions
in strategic areas such as those involved in planning and budgeting. The
shortcomings just mentioned can be overcome, and many important factors
in capital investment decisions can be taken into account in a model that
truly represents corporate operations and can truly be solved with the
computer technology of today.
The approach to capital budgeting that I shall describe in this article is
a new one. It has, however, been tested in a variety of situations, and I am
convinced that it can be used profitably by a great many companies.
MAKING THE ANALYSIS
In explaining the new technique it will be helpful to refer to a company
example:
Let us assume that the company is in one of the process industries. The plant
in question is physically adequate but technologically obsolete. Management is
faced with the decision whether to invest capital in modernizing the plant or to
scrap it for salvage and tax advantages.
If management modernizes the plant, it can take advantage of existing offsite
facilities with several possibilities for expansion. If, on the other hand, the plant
is scrapped, alternative sources of supply must be utilized. These in turn may
MATHEMATICAL MODELS IN CAPITAL BUDGETING 333
involve additional capital investment in modernization, expansion, new trans
portation, storage facilities, and so on. Alternative possibilities may include
purchasing, exchange, and processing agreements with other manufacturers.
In real life there are often thousands of possible combinations of manu
facturing, transportation, and marketing investments in a case of this kind.
In addition, the demographic changes in today's economy present a very
real possibility of gross changes in the pattern of demand over as short a
period as ten years, so that the operating configuration that is ideal for
today's market may, in fact, be highly undesirable just a few years from
now. In too many cases the existence of a problem of this complexity leads
to total disagreement among members of top management. The disagree
ment is usually resolved by a compromise solution acceptable in the short
term, but without enough regard for longterm corporate interests.
To meet the need, it is often possible to construct a model of the com
pany's activities that permits a rational decisionmaking mechanism. The
operations of our process plant, for instance, may be fairly represented as
including manufacture, distribution, and sales. Each of these operations
has cost components associated with it, which components in turn may be
divided into capital costs, fixed operating costs, and variable operating costs.
The system may be totally represented by a simple mathematical model,
effectively balancing supply and demand on a detailed basis throughout the
company's operations. The structure of such an operation might be devel
oped along the lines indicated in the Appendix .
A system of this type is capable of being "optimized"; that is, an operat
ing procedure consisting essentially of an allocation of production and
supply can be found to minimize the total cost to the enterprise. This has
long been known, and the method has been widely applied to operating
problems such as the minimization of transportation costs. 1 It is possible,
however, to use the same simple computing method as a guide in making
investment decisions.
FUNCTION OF CAPITAL
In order to do this, we must recognize that the first decision to be made
involves a definition of the function of capital in the company. What is the
capital to do? To an investment banker, the function of capital is to make
money, but not necessarily so for the chemical manufacturer. So far as the
management is concerned, so far as the mode of operation is concerned,
1 See, for example, Alexander Henderson and Robert Schlaifer, "Mathematical Pro
gramming: Better Information for Better Decision Making." reprinted in this book on
page 53 ; C. W. Churchman, R. L. Ackoff, and E. L. Arnoff, Introduction to Opera
tions Research (New York, John Wiley & Sons, Inc., 1957).
334
FACILITIES PLANNING
and so far as the investment policy is concerned, the function of a company
is to get raw material manufactured into a finished product, distribute that
product, and sell it to the customer. These are the operations ; these are the
objectives, the things to do.
The junction of capital, then, is to permit management to meet the
manufacturing and marketing objectives more efficiently. A measure of this
efficiency is a lowering of the operating cost to the company. This means
that the costs used in the model can be stated without consideration of any
capital invested. They are, in fact, true operating costs for labor, raw
materials, utilities, and so on. There are no charges for depreciation, taxes,
or return on investment at this stage of manipulation.
PRODUCTION COSTS
We come now to a second important step in the construction of our
model, or rather in the philosophy on which the model is based. This has
to do with the treatment of production costs.
In many operations the unit cost of production decreases with the volume
of the operation, and is generally taken as a variable operating cost. We
need not consider it that way. We can define the true operating cost as the
total cost of operating at full capacity, divided by the production at full
capacity. Any cost higher than this is not a cost of production; it is a cost
of Azewproduction.
These two types of costs may be defined by a curve as shown in Exhibit
I. (This is a very simple illustration, of course. In many cases the relation
EXHIBIT I. PRODUCTION AND NONPRODUCTION COSTS
•• UNIT PRODUCTION COST
 UNIT COST OF NONPRODUCTION
TOTAL PRODUCTION
MATHEMATICAL MODELS IN CAPITAL BUDGETING 335
ships might not be nearly so linear as portrayed here.) Note that the unit
cost of nonproduction is zero at full capacity and 100% of the production
cost when the machines are all stopped.
This treatment of costs is satisfying in that it places a penalty on failure
to use a facility, the use of which would increase efficiency. It has a further
mathematical advantage: whereas cost as usually defined is a variable, the
two costs as now defined are true constants without approximation. This
advantage promises an enormous simplification of the computing. The
problem may now be solved and the optimal solution found by means of
the simple calculation given in the Appendix. The steps are as follows:
( 1 ) Find the cost associated with various modes of operation, taking into
account available (or possibly available) manufacturing, transportation, and
other facilities.
(2) Compare the differences in the operating costs with the differences in the
capital investments required to achieve those costs.
(3) If the relationship is such that the savings in operating costs are at least
as much as the return required on the capital investment, then the investment
should be made. Preferably the return should be calculated on a discounted
cashflow basis 2 so that the differences in economic life, in extent of obsoles
cence, and in book value may be fully taken into account.
PRACTICAL ALTERNATIVES
In order to make a sound longterm decision, the problem should be
studied at several points in time, the restrictions being assumed for as long
a time period as possible. If we are extremely fortunate, we will find that
the best modes of operation as calculated for different periods of time are
compatible; that is, we will find that the best mode of operation for today
can be logically expanded into the best mode of operation for several years
from now. Unfortunately, in real cases this may not happen. For example:
In the case described in Exhibit II, it was found that a particular plant should
be modernized and operated for three years, should then be placed on a
standby basis and not enter the distribution system, and should then be re
opened and expanded.
Such a solution, although mathematically valid, is totally impractical. In a
case like this, one can define the practical strategies that most nearly fit the
ideal. In the example given there are, in fact, three courses of action open to
the management:
( 1 ) The management may modernize the plant, keep it open, and adapt the
distribution system to its existence in the middle years.
2 For a discussion of the logic, principles, and application of the discountedcash
flow approach, see Ray I. Reul, "Profitability Index for Investments," HBR July
August 1957, p. 116.
336 FACILITIES PLANNING
(2) It may modernize and operate the plant in the early term, then close it,
and operate without it in the last period of years.
(3) It may shut the plant immediately and operate without it throughout the
entire period.
The basic model may be resolved to meet these practical requirements. The
three modes of operation may then be compared by again taking differences in
capital and in operating costs over the entire time period and discounting back
to present value. The minimum cost alternative can thus be found and accepted.
EXHIBIT II. CASE IN PROGRAMMING A CAPITAL INVESTMENT
A problem arose because of the existence of a manufacturing plant which, although
physically adequate, was technologically obsolete and unable to supply the quality of
products required in today's market. The plant in question was rather small compared
to those then being built. Since it was argued that a plant of this size was at an in
herent disadvantage for economic operation, there was considerable managerial con
troversy over the proper course of action — whether to modernize the operation by
construction of better facilities or to scrap the existing plant and supply the area in
volved from facilities elsewhere, e.g., plants in adjacent states.
POSSIBILITIES CONSIDERED
Investigation of the managerial decisions to be made disclosed that the problem
involved the choice of the best combination of facilities shown in Figure A.
The Jonesboro Plant was the one under consideration. It could have been closed,
modernized at present capacity, or modernized and expanded in various degrees. The
Smithville and Johnstown refineries could also have been modernized at present
capacity; other possibilities for them were moderate expansion (enough to remove the
bottleneck in existing facilities) or major expansion. Anderson, Boylstown, Charles
town, and Davis represented important market areas each of which, together with
Jonesboro, could have acted as a terminal for secondary distribution of the product if
transportation facilities were installed (see the dottedline route) to supplement those
existing from Smithville to Anderson and the available barging from Johnstown to
Davis. Additional flexibility in operation was provided by the possibility of executing
agreements for bulk purchase and sales, product exchange, and raw material process
ing with other manufacturers in the area.
In all, considering the various combinations of manufacturing and transportation
facilities, plus agreements with other manufacturers, there were some hundreds of
operating policies to be considered at any given time. Furthermore, the longterm
consequences of immediate decisions were of major importance.
Available market forecasts indicated that various means of upgrading product
quality would be required. Also, available forecasts of quantity demand, considered to
be of acceptable accuracy for the first five years and of lesser accuracy for the next
five, indicated that the demand pattern would shift considerably during the period, so
that care had to be taken not to penalize operations at some later "date so as to make
an immediate good showing.
EXPRESSION OF PROBLEM
Analysis disclosed that, for any assumed management decision as to selection of
facilities, the operating problem could be expressed as that of optimizing manufacture
and distribution of products in a system consisting of approximately 200 destinations
and 25 origins (plants, terminals, and points of exchange or purchase). It could
MATHEMATICAL MODELS IN CAPITAL BUDGETING
337
338 FACILITIES PLANNING
therefore be defined as a transportation problem of the type given in the Appendix.
In t the study of the costs involved, some peculiarities became evident. Rates for
trucking, barging, or shipment were linear (i.e., the cost per unit was the same, re
gardless of the volume shipped). But pipeline costs on a unit basis were nonlinear and
varied in a manner which seemed to be dependent on the method of financing.
The major cost component — that of manufacture — was said to be a "step function."
This meant that if a basic volume were produced at a certain unit cost, an additional
"incremental" volume could then be produced at a lower unit cost per volume for that
increment only, that a still lower unit cost then applied over a third range of volume,
and so on. By appropriate devices these factors were introduced into the model, which
became subject to optimization.
In constructing the cost tables, it was early apparent that the definition of costs
could easily be varied in such a way as to favor certain types of investment and
certain methods of operation. Two particular pitfalls existed:
(1) The divisions of the company — e.g., manufacturing, transportation, market
ing — operated virtually autonomously, and internal accounting was on the basis of
"transfer prices." Since these prices could include divisional overhead and profit, it
was not appropriate to try to minimize them for the company as a whole.
(2) The existing investment was subject to various degrees of amortization, and
it was known that new capital would be required at various times throughout the
period studied. Accordingly, incautious use of capital investment charges could have
led to fallacious answers.
To eliminate these dangers, all costs were constructed on the basis of true operating
costs, with such components as interest, insurance, ad valorem taxes, and depreciation
omitted.
Particular combinations of facilities were selected on the basis of an approximate
balance of supply and demand throughout the period to be studied. The years at the
beginning, end, and middle of the period were first studied, and the optimal mode of
operation and distribution determined for each practical combination of manufactur
ing, transportation, and marketing facilities. This resulted in a series of cases for
each of the key years. In order to compare these cases, decreases in operating costs
were matched against increases in capital requirements.
USE OF MODEL TECHNIQUE
This step is a key point of the model technique. Recognizing that there is not an
infinite pool of capital available for investment, we adopt the principal that the
junction of capital in industry is not only to earn a return but more specifically to
earn the return by decreasing operating expenses. For example, typical results might
be as listed in Figure B.
New investment
(millions of dollars)
$10.0
12.5
16.0.
18.0
20.0
Here Case # 4, which has the lowest operating cost, is the one with which to start
comparisons. Case # 5 is clearly not so good as # 4 since both its capital requirement
and operating costs are greater. In Cases #1, #2, and #3 capital requirement
is decreased at the expense of increased operating cost, as shown in Figure C.
Figure B
Case
Operating cost
(millions of dollars)
#1
$53.0
52.0
50.4
50.0
51.0
MATHEMATICAL MODELS IN CAPITAL BUDGETING 339
Figure C
Increase in cost Decrease in capital
compared to compared to
Case # 4 Case # 4
Case (millions of dollars) (millions of dollars) Ratio
#1 $3.0 $8.0 0.375:1
#2 2.0 5.5 0.364:1
#3 0.4 2.0 0.200:1
These increases in operating cost may be looked on as the cost of the decreased
amount of capital. Obviously the best of these is Case # 3, where $2 million of
capital is obtained at an increased operating cost of only $400 thousand per year.
That is, Case # 3 is the best of the alternative cases.
To determine whether it is absolutely preferable requires the establishment of
bench marks for comparison. The bench mark is the acceptable return in this type of
investment as defined by management. What we want to take into account is the
lessthaninfinite pool of money available, and the existence within the company of
competitive opportunities for investment. The bench mark is properly set by an
evaluation on a discountedcashflow basis by type of investment. Thus if a dis
counted return of 14% over a 25year economic life is set, the ratio of the decrease
in capital divided by the increase in cost is 0.255, or greater than the 0.200 for Case
# 3. Case # 3 is therefore preferable and should be chosen over Case # 4.
Let me put this in a slightly different way. In Case # 3 the company needs to in
vest $2,000,000 less capital than in Case # 4. At a discounted return of 14%, the
company could earn $510,000 on this freed capital ($2,000,000 X 0.255). The price
of earning this sum is $400,000, which is the increase in cost of Case # 3. Thus, the
company is gaining $110,000 which it would not have if it chose Case # 4.
SOLUTION
Proceeding in this fashion, optimal operating situations in our case example were
chosen for the key years at the ends and the middle of the tenyear period. The
results showed a curious effect. Because of the shifts in supply and demand during
the period, the plant was scheduled to produce at base capacity initially but did not
enter the solution (i.e., was closed) at the midperiod; it came in again for produc
tion at expanded capacity in the solution for the end period. That is, the computa
tions showed that it would be desirable first to modernize the plant, then to close
and "mothball" it, and finally to reopen and expand it. Investigation showed a three
year period over which the plant should be on standby.
Such a procedure was, of course, unthinkable from an administrative standpoint.
Although a plant may be "mothballed," people cannot be. Therefore, the practical
alternatives were examined. The company could:
(1) Modernize the plant initially and operate until nonoptimal, then close for
good and operate with the nextbest arrangement of facilities during the final
period.
(2) Modernize initially and operate over the entire period, using the nextbest
method of operation during the middle period.
(3) Close the plant immediately and operate according to the nextbest method
during the nonoptimal periods at the beginning and end of the time studied.
To make this choice, the possible investment schedules for each were determined
and the model manipulated to find optimal operating costs for each alternative over
the entire period of the study. The differences in operating costs and investments were
then discounted over the entire period to enable comparison, and a choice was made as
before. It appeared that the second alternative was best, and the third alternative
next best. The advantage of the second over the third was that total investment was
16% lower, and the return, after discounting, higher.
340 FACILITIES PLANNING
TREE OF CAPITAL
The basic model gives a "rough cut" solution for a longterm pattern of
investment designed to fit the changing conditions envisioned in the plant.
The job is not yet finished, however. The model thus far will have con
sidered broad operations without regard to the economic desirability of
individual parts. To include this factor we proceed to a second stage of
model building. From the results of our computations at this stage we
construct a "tree of capital" which diagrams the relation of the various
parts of the enterprise in the manufacturingdistributionselling complex.
The structural relationships in the tree do not depend on company
organization. They are, in fact, inputoutput relations, the facilities at each
level being supplied by the facilities at the next level below. The relation
ships also define two flows: the flow of product from bottom to top (like
sap) and the flow of cash from top to bottom (like rain on the leaves).
Initially in the computations, no portion of the enterprise is permitted to
earn anything; the cost of supplying the highest level is calculated with the
costs that were used in the model. The cash generated is at first assumed to
be associated with the highest level of the tree. The various facilities at this
level may have very different values for the capital invested and the cash
generated. We proceed as follows:
As a first step, we let the cash accrue only to each facility at the highest level
and calculate the return on the investment at that point. Some one point will
have the lowest rate of return. We now assume that this is a benchmark rate of
return for this type of investment, and we permit every facility at this level to
earn at this rate.
All cash beyond the amount to be earned is reflected to a pool in a second
level facility and associated with that. The same type of analysis is performed
at this level for the operations of the system.
The process is repeated at each of the various levels until eventually we come
to the bottom of the tree and have a cash pool sufficient to pay off company
expenses, overhead, and so on. If there is not enough money to do this, it is
evident that some of the bench marks have been set improperly.
If it appears that some of the bench marks are in error, we eliminate the
marginally productive facilities at various levels and reallocate our cash until
eventually the system is in balance. If any major facility is eliminated, the basic
model must be changed to reflect this fact and a new solution found to correct
the structure of the tree.
FLEXIBILITY ADDED
The model thus gives us flexibility in a most important respect, viz., a
balanced amount of parasitic capital is allowed to appear in the company.
MATHEMATICAL MODELS IN CAPITAL BUDGETING 341
The value of this can be readily appreciated if we recall that many com
panies have a flat rule that a new facility costing, say, a million dollars
must return at the appropriate rate on that million dollars, plus a certain
percentage to go into company overhead. This rule does not allow for the
attractive marginal investment which indirectly contributes to the total
company. Also, in many cases representation in an area is desirable or even
necessary for effective company operations.
There is no general rule, nor should there be any general rule, as to the
extent to which facilities must support the enterprise. If, in fact, the invest
ment is in balance on the tree of capital, we have a healthy enterprise.
APPORTIONING FUNDS
As a further application of the modelbuilding technique, we can devise
another tree which represents not flow of goods but flow of decisions. This
flow should closely resemble the scheme indicated by the company's organi
zational chart. We find here various levels of decision making, with a
narrower range of activities and interests as we go to the lower levels.
Requests for investment originate at all levels and are passed up to higher
levels. At the very highest level top management is faced with the problem
of budget allocation, and probably for the first time consideration is given
to the question of return to the whole company rather than to a division or
department.
Since in the usual case requests for funds exceed the funds available,
there is a definite need for an optimum allocation. One procedure here is to
begin at the lowest level of the enterprise, incorporate the planned facility
into the company model, and determine the prospective return on the
capital required. This should be done at the lowest level for each invest
ment proposal, with those reporting to a common point on the second level
being grouped together.
The return functions may now be compared. For example:
Suppose that management is considering one proposal to build Unit A and
another to build Unit B, both in the same area of decisionmaking responsibility.
Suppose further that if Units A and B are given unlimited funds, the estimated
rates of return will be as shown in Exhibit III.
Now, the fact that the return for Unit A is higher than that for Unit B at
all levels does not imply that all available capital should be invested in
342 FACILITIES PLANNING
EXHIBIT III. PROJECTED RATES OF RETURN FOR TWO PROJECTS
'mmmmmfK.
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I
TOTAL INVESTMENT
Unit A. Instead, the initial capital should go into Unit A until the point
where the incremental return falls below the initial rate of return for Unit B
(as happens in the last half of the Unit A curve in Exhibit III). At this
point some funds should be diverted to Unit B in order to maximize the
return to the enterprise as a whole.
In fact, for any given level of investment there is an optimal allocation
between the two units. With the necessary data at hand, this result could be
expressed in curves as shown in Exhibit IV. We see from this chart that all
EXHIBIT IV. ALLOCATING FUNDS BETWEEN COMPETING PROJECTS
$1 $2 $3
LEVELS OF INVESTMENT (MILLIONS OF DOLLARS)
MATHEMATICAL MODELS IN CAPITAL BUDGETING 343
of the first million dollars of investment should go into Unit A; in the case
of the second million, however, somewhere around $950,000 should go
into A and $50,000 into B; in the case of the third million, about $900,000
should go into A and about $100,000 into B; but after that A's share rises
again.
Such a set of curves can be constructed for three or more competing
investment opportunities. The result of them all, the output from the
analysis of allocation at the lowest level, is the input to a similar analysis
for the second level. At the second level, the returnoninvestment curves
define another set of enterprises competing for capital.
Such an analysis may be carried on step by step to top management, each
decisionmaking level being in turn supplied with a plan for allocation of its
funds.
IMPROVING DATA
One caution should be noted. No decision is better than the data avail
able or the assumptions made. This is true for decisions based on a model
as well as for those reached by any other means. The model approach,
however, has two great virtues:
( 1 ) Assumptions involved in defining a problem are necessarily made explicit
when formulated mathematically, although this is not necessarily done in many
other approaches.
(2) It is possible in many mathematical models to test the sensitivity of the
result to the input data. Such sensitivity analysis may then indicate the areas
where additional work in defining economic or operating quantities may be of
importance.
APPLICATIONS
How broadly applicable is the kind of analysis I have described? As we
have seen, one kind of problem that can be solved is that presented in
Exhibit II. But other kinds of problems, too, lend themselves to this type
of solution. Let us look at some of them.
CHOICE OF PROCESS
One such problem is that which production executives sometimes face
when a company or plant is expanding. To illustrate, here is a situation that
has recently come up:
A manufacturer is faced with a sales forecast which makes it necessary for
him to expand facilities within a definite period of time. He has a proven process
which may be used and which also is used by his competitor. He also has small
scale experimental information on two modifications which make patentable
344 FACILITIES PLANNING
distinctions in the product and affect the economics of production. It is not
certain in advance, however, that either of these modifications will apply favor
ably in largescale production.
It has been found possible to design plants in various ways. Different designs
are affected favorably or unfavorably by the existence in largescale manu
facture of effects observed in smallscale experimentation. A model of the
system has been constructed and the theory of games applied to find the solution.
In solving the game, with quantities reflecting the advantage or disadvantage
of the proposed process relative to the existent process, it has been found that
there are several equally good alternatives. These represent the minimum return
to be expected, regardless of what happens in largescale work. The solution
thus shows what is the least to expect in the face of uncertainty, with a high
probability that a greater return will be achieved.
MARKETING ANALYSES
Other kinds of problems arise in the distribution system. Here are two
illustrations, both from actual experience:
C A problem in capital budgeting arose and was solved as a consequence of
studying a large geographic area consuming 6% of a company's products but
supplying only 3% of its marketing profit. The problem was to —
(a) Find the reasons for the unprofitability.
(b) Determine if the region should be abandoned as a market or, if not, what
steps should be taken to make it profitable.
The disproportion of profit to sales was found to be caused partly by the high
cost of product and partly by improper marketing. The major sources of supply
were plants which were among the company's oldest facilities with a high pro
duction cost relative to the company's other plants. This situation was aggra
vated by highcost purchases in the area; and, in addition, high transportation
costs prevailed.
The direct costs of marketing were comparable with those applying elsewhere,
but unallocated expenses were lower. Thus, the region had a problem of in
adequate margins.
So a mathematical model of the manufacturing and distribution system was
constructed, and optimal operation under a variety of assumptions was studied.
The results showed that the region could best be supplied by keeping one plant
in its existing state, expanding another plant by approximately 25% of its
capacity, and upgrading the product at the remaining plants while maintaining
their thencurrent capacity. At the projected level for unallocated marketing
expense, the overall return to the system was less than the company's target
but compared favorably with the company's achievement. Management saw
that if these expenses could be reduced below the projected level, a very favor
able overall return could be achieved.
C A major manufacturer of consumer goods is faced with the problem of
MATHEMATICAL MODELS IN CAPITAL BUDGETING 345
deciding what to do about his distribution system. He currently distributes
through a system of franchised distributorships, each of which independently
controls field warehouse inventory levels, the intensity of marketing effort at
the retail level, and to some extent the intensity of advertising in its locality.
In the face of the projected expansion of sales during the foreseeable future, it
is apparent that the existing facilities of the distributors will be inadequate.
A study of this problem has incorporated the factors of plant and field in
ventory, the freight costs required, distributor and dealer markups, and an
elaborate forecast of the economic parameters or variables which affect sales.
The data produced enable the company to make a decision between (a) systems
based on distributorships, regional warehouses, and freight forwarding and
(b) directtodealer systems.
SPREADING INVESTMENTS
Still another kind of problem that lends itself to the analytical approach
which I have described has to do with the composition of an investment
portfolio. For instance:
A large investment company has used the modelbuilding technique to study
the problem of constructing a portfolio of investments. At the same time that it
has considered the factors of relative return and relative risk in various invest
ment opportunities, it has incorporated not only management's desire for
diversification but also the competitive necessity of producing both income in
the form of dividends and capital gains in the form of appreciation.
The foregoing cases indicate something of the breadth of applicability of
the model manipulation concept. The solutions found have already, in some
cases, been applied for a sufficient period of time to show that the results
predicted by the model can be achieved.
CONCLUSION
It must be emphasized that use of the model or of any of the mathematical
techniques of the operations researcher does not imply management by
computer. The mathematical model itself is a tool of management rather
than a replacement for management. The factors to be considered in con
struction of the model are those which are and must be taken into account
in any thorough decisionmaking process. These would include, for ex
ample :
A description of the potential market over a given period of time in terms
of the probable demand pattern for the entire area.
The possible points for production distribution both already in existence and
to be contemplated in planning.
Typical production restrictions — for instance, whether or not a given plant
346 FACILITIES PLANNING
may be considered as being tailormade to produce one product or to be capable
of flexible output.
An estimate of the operation and distribution costs, classified as capital
and/or noncapital charges and described preferably as a function of the volume
of production. (These costs should include actual records as well as expenses
projected and studies for new plants.)
Any appropriate managerial restrictions such as a policy of constancy of
employment; a decision on whether or not overtime or layoff dollars may be
used; a decision on whether or not plant capacity must be fully or almost fully
utilized at all times; a rule on whether or not labor productivity or capital invest
ment is to be considered a prime objective; a preference for stating return in
absolute dollars rather than as a rate on investment, or vice versa; a decision on
whether or not control of a market is considered to have an economic value; or
a policy committing the company to uniform exploitation of a broad area.
These quantities constitute the body of facts and assumptions on which
the decision must be made.
MORE THOROUGH ANALYSIS
The concept of model building outlined in this article has important ad
vantages for modern business. For one thing, it enables the executive to
probe more deeply and more thoroughly into the factors that affect a
decision. Characteristically, managerial problems contain many more vari
ables than restrictions, so that in a real case thousands of solutions may
exist. The function of management lies in defining realistic assumptions and
practical operating conditions. The computer can then perform its function
of taking these restrictions and performing the detailed labor of investi
gating their consequences for the solution.
The output from the manipulation of the model is then a detailed plan of
action which is only tentative over the period of study. Indeed, the solution
itself may suggest to management introduction of modifications to recognize
the effects of additional factors, changes in estimates, possibilities of diversi
fication, and so on. In fact, the existence of the model, and of computers
capable of dealing with the model, enables management to make an
exhaustive study of possibilities rather than a comparison of some of the
more obvious cases.
Of course, construction and manipulation of a model are not to be under
taken lightly, or in the expectation of achieving results overnight. Costwise,
much depends on the excellence of existing data and the magnitude and
complexity of the job to be done; but it is fair to say that data collection
and refinement, the incorporation of economic analyses and forecasts, and
mathematical analysis and computer operation may be a long and costly
process, with costs running from $30,000 to $100,000 in some cases. It
MATHEMATICAL MODELS IN CAPITAL BUDGETING 347
should be noted, however, that this figure is largely a "setup cost" and that
the model, once constructed, may be kept current at a comparatively low
expense so that successive applications can be made relatively cheaply.
To sum up, the use of mathematical models can supply management
with a tool for decision making at virtually all levels, from daily operations
to budget allocation and longterm capital investment programs. The full
potential of the method has not yet been explored, but it is apparent that
this technique can be of great aid in managerial decisions.
Appendix
CONSTRUCTION AND USE OF A MODEL
In this article, and in the literature of mathematical programming generally, the
terms "model" and "mathematical model" are used rather frequently. The usual
meaning is that of a group of equations which purport to describe the problem
under study in such fashion that all proper considerations are explicitly stated, so
that the solution of the system of equations is in fact the solution of the real problem.
By way of illustration, let us construct a model for a relatively simple problem in
distribution.
MATHEMATICAL STATEMENTS
Consider the problem of distributing a commodity in a system having balanced
supply and demand, in which ten customers, 1 to 10, are supplied from four points,
A, B, C, and D. In the broadest form of the case, any customer can be supplied from
any one of the four points, and the problem is to determine that policy of distribution
which minimizes the cost of transportation for the entire system. Data typical of
such a situation might be as given in Table A. Here customer demands and ware
house supplies are shown under the columns and to the right of the rows respectively.
The numbers in the body of the table represent the unit cost of supplying each
customer from each warehouse.
Objectives & Restrictions. The table is itself, in a sense, a model of the problem,
a better and more generally expressive model can be constructed by a mathematical
statement of objectives and restrictions — the objectives being statements of what is to
be accomplished, and the restrictions being statements of all considerations which
enter into the policy decision. To express these in equation form, the variable in the
solution is defined as being the units of the commodity being supplied from each
warehouse to each customer.
348
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MATHEMATICAL MODELS IN CAPITAL BUDGETING 349
In Table B, Xi indicates the (unknown) number of units to be supplied to Cus
tomer 1 from Warehouse A; x 36 the number of units from Warehouse D to
TABLE B.
THE VARIABLES
Customer
Warehouse
1
2
3
4
5
6
7
8
9
10
A
Xi
x 2
x 3
x 4
x 5
x 6
x 7
x 8
x 9
X10
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Xn
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Xl3
X14
X15
Xie
Xl7
Xis
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X20
C
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X23
X24
X25
X26
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X28
X20
X30
D
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X32
X33
X34
X35
X36
X37
XsR
X39
Xto
Customer 6; and so forth. This being so, any shipping schedule may be expressed by
giving the numerical values associated with each of the x's. One such schedule
might be :
Xi =2,114 x 26 = 2,371
x 2 = 1,797 x 27 = 612
x 3 — 532 X37 = 311
x 18 = 270 x 38 = 840
x 14 = 2,032 X39 = 953
Xi5 = 2,760 x 10 = 1,547
Xi« = 209
All other x's would be zero, representing shipping combinations not used.
Similarly, the cost associated with any shipping schedule is given by the sum of all
products of the values of the variables and the unit costs given in Table A. In
equation form:
Cost = 4.41 Xi + 4.60 x 2 + 1.50 x 3 + • • • + 6.55 x.o
This equation is the objective function, and the objective is to minimize this cost, while
meeting all restrictions. The restriction in this simple problem are easily stated: at all
points supply and demand must balance. We write one group of equations stating that,
for each warehouse, the sum of the variables associated with it (shipments out) must
equal the amount available at that warehouse; another group of equations will simi
larly state that the sum of variables associated with each destination (shipments in)
must equal the demand at that point. These restrictions are shown in Table C.
All of these equations together represent the model, and the problem as given is
that of minimizing the objective function subject to the restrictions. The supply and
demand restrictions are the only ones having meaning in the simple problem stated
here. In real problems conditions of policy, legal requirements, capacity, physical or
chemical properties of materials, and so on may be incorporated.
SOLUTION
At first glance the technique of model construction may seem unnecessarily compli
cated, but it is not really so. The characteristic of management problems is that, when
so stated, the number of variables involved is much greater than the number of
equations required to express the restrictions. This is equivalent to the statement —
certainly no news to most executives — that there are, in general, many ways to operate
in a real system. Thus, even in the simple problem posed, there are many million of
shipping schedules that could conceivably be devised. And in any real case the least
cost solution may be by no means obvious, For the data as presented above, the
optimum solution and the costs of departure therefrom are shown in Table D.
350
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In this table, the numbers in bold face represent quantities to be shipped from the
indicated warehouse to the indicated customer. Thus Warehouse A is to supply 940
units to Customer 1; 2,580 to Customer 6; and 923 to Customer 7. The remaining
table entries in all the warehousecustomer combinations not called for in the optimal
solution represent penalty costs. Thus the 0.21 associated with A2 indicates that the
MATHEMATICAL MODELS IN CAPITAL BUDGETING 351
total cost of shipment will be increased by a minimum of 0.21 for each unit shipped
from Warehouse A to Customer 2. The fact that all routes other than those called for
in the solution have positive costs associated with them indicates that this is a least
cost, or optimal, solution.
There are several surprising features to this solution. It will be noted, for example,
that no one of the nine lowestcost routes (A3, A4, A9, B3, C3, C4, C7, C9,
and C10) is actually used. Furthermore, of the 10 customers, only Customer 1 is
totally supplied by the leastcost route available to him. Thus Customer 3 is totally
supplied from Warehouse D, although the unit costs are lower from any one of the
other warehouses; the apparent saving of 1.71 to be achieved by supplying Customer
3 from Warehouse A rather than from Warehouse D becomes an overall loss of 0.55
for the system. Yet it is to be emphasized that the solution as given is, overall, that
of minimum cost, as may be seen by computing the cost of alternative solutions.
OPTIMIZATION PROCEDURE
The construction of a mathematical model for such a simple problem as that pre
sented by the assignment of a shipping schedule to the four warehouses and ten cus
tomers seems like an elaborate and formal procedure of little real value. However, if
in large systems there is something to be gained by such an approach, the formal pro
cedure may become more appealing. It will be readily seen that, even for the small
problem earlier described, many shipping schedules may be devised; and in a real in
dustrial situation, the magnitude may be such that literally billions of solutions to the
problem may exist. Mathematical programming makes the claim that out of these
billions of solutions the optimal solution can be found — the solution which is not
only a good or even very good method of operation, but which is the best of all
possible solutions. This claim may be verified with mathematics no more advanced
than addition and subtraction.
Feasible Solution. The first step in optimizing the system is to obtain a "feasible"
solution — one that meets the restraints of the problem without regard to cost. In the
case of the problem of Table A, this may be done systematically in simple fashion.
Let the first customer have his demand filled as fully as possible from the first ware
house. If there is more at the warehouse than is needed to supply this first demand,
the excess is supplied to the second customer; and so on until the warehouse supply
is exhausted.
Generally the customer exhausting the warehouse will not have his demand fully
met, in which case the difference is filled from the next warehouse; then the next cus
tomer is supplied. The process is continued until all demands are met and all supplies
exhausted. For the illustrative problem this gives the shipping schedule shown in
Table E.
TABLE E. SHIPPING SCHEDULE
Customer
Warehouse 1
2 3 4
5 6
7
8
9
10
A 2,114
B
C
D
1,797 532
270 2,032
2,760 209
2,371
612
311
840
953
1,547
This schedule, represents a managerial policy employing 13 of the 40 warehouse
customer combinations possible. Any simple departure from this policy involves use of
one of the 27 unused warehousecustomer combinations. Whether or not such de
partures are good depends on the cost difference resulting from such a departure.
352 FACILITIES PLANNING
Departures. To evaluate a departure from this policy, we reason as follows. The
designated solution represents a balance in supply and demand not only overall but in
each row or column separately and independently. A change in the value of any
quantity must be compensated for by an equal and opposite change in another quantity
in the same row and another in the same column. These two changes demand in turn
two new changes, in a column and a row respectively; and so on.
The process terminates only when it is possible to satisfy both a row change and a
column change simultaneously by finding a quantity which can be altered at the inter
section of the row and the column. Further, since negative quantities cannot be intro
duced into the solution, the changes to compensate for introduction of a new variable
— that is, employment of a hitherto unused warehousecustomer combination — can be
made only by decreasing quantities in the solution.
For example, to depart from the policy defined by the solution given in Table E by
supplying Customer 1 from Warehouse B demands that the supply to Customer 1
from Warehouse A be decreased by a corresponding amount, since otherwise Cus
tomer 1 will be oversupplied. At the same time, the supply to one of Customers 3, 4,
5, or 6 must be depleted to avoid demanding more than the total supply from Ware
house B. Of these, it is seen that if Customer 3 is the one whose supply from Ware
house B is lowered, then increasing the supply to Customer 3 from Warehouse A com
pensates for both changes so that the solution need not be further disturbed.
Similarly, it may be seen that to depart from the indicated policy by supplying Cus
tomer 2 from Warehouse D (combination D2) involves compensating decreases in
A2, B3, C6, and D7, and equal increases in A3, B6, and C7. In general, the
compensating changes involved in utilizing a new route will be traced out by such a
series of moves, and there will be only one such series of changes possible.
The economic consequences of such a departure from the assumed policy, in terms
of the total change in cost of the shipping schedule, can be found by adding the unit
cost associated with any route where the assignment is increased, and subtracting the
unit cost if the assignment is decreased. Thus the effect on the first feasible solution
of use of the combination Bl (supplying Customer 1 from Warehouse B) is given
by the figures:
5.56  4.41 + 1.50  2.38 = 0.27
and use of the combination D2 (supplying Customer 2 from Warehouse D) changes
the solution value by the following amount:
6.63  4.60 + 1.50  2.38 + 4.96  3.65 + 3.03  5.75 =  0.26
Thus the first change increases the total cost, and is therefore unattractive; the sec
ond decreases the cost and might be incorporated into the solution. In this manner
every possible change may be evaluated by calculating the associated change in cost
for unit utilization The solution may then be improved by incorporating the route hav
ing the largest associated saving, and doing this to the fullest possible extent.
The extent of incorporation (that is, the number of units to be assigned to this
route) may in turn be judged from the compensating changes made in evaluating the
cost of unit utilization. Those changes which increase in magnitude need not be con
sidered, since there is no upper limit on their magnitudes. Those which decrease in
magnitude, however, may not go below zero, so that the largest change that may be
made is equal to the smallest value of the group. When this value is determined, all
the routes affected may be changed by the appropriate amount. This constitutes a new
solution, corresponding to another policy of operation.
All possible departures from this policy may now be evaluated and the process
repeated until a solution is obtained from which all changes are unprofitable. Such a
solution is, of course, the optimum one.
MATHEMATICAL MODELS IN CAPITAL BUDGETING
353
Recapitulation. This process of evaluating departures from the first feasible solu
tion has been reviewed in perhaps tedious detail in order to demonstrate two facts:
( 1 ) The manipulation of the model is logical at an elementary level.
(2) The result of the manipulation is truly optimal; it is the best possible solu
tion for the problem as stated.
Summarizing, it may be stated that the optimization procedure consists of certain
definite steps:
(1) A model is constructed, consisting of a group of equations which explicitly
describe all restrictive relations in the system.
(2) A feasible solution, or possible operational policy, is found without regard
to value.
(3) All possible departures from the policy are identified and evaluated for their
effects on the value of the policy.
(4) The departure with the greatest unit value is incorporated to the fullest
possible extent, making a new feasible solution.
(5) Steps 2, 3, and 4 are repeated until a solution is found which cannot be
improved on.
MORE COMPLICATED QUESTIONS
The illustrative examples just presented have been simple in conception and repre
sent what may be felt to be an idealized state. However, various degrees of com
plexity may be introduced.
Supply Bottleneck. For example, consider another problem of four sources and ten
destinations, with the appropriate quantities as given in Table F, and the solution as
shown in Table G.
TABLE F.
TRANSPORTATION COST PER UNIT TO DESTINATION
A vail
Destination able
Source Di
j rum
D, D 3 D,, Ds D 6 D 7 Ds D 9 D 10 source
Ri
R 2
R 3
R*
Demand
at desti
1.45
1.67
1.48
0.85
1.15
1.45
1.20
0.65
0.21
0.50
0.30
0.10
2.54
2.80
2.59
2.00
1.98
2.15
2.00
1.25
0.49
0.80
0.60
0.38
1.00
1.26
1.10
0.89
0.36
0.75
0.55
0.25
2.35
2.75
2.40
1.75
3.00
3.25
3.00
2.25
6,500
2,000
4,800
4,800
nation
2,000 2,400 1,500 1,300 1,300 2,100 1,800 1,700
2,200
1,800
TABLE G. SOLUTION
TO SECOND PROBLEM
Destination
Source
Di D 2 D 3 D h
D 5 D 6 Di
Ds
D 9
D 10
Ri
R 2
R 3
R 4
0.04 0.02 1,500 0.98
1,400 0.06 0.03 0.02
600 2,400 0.02 1,300
0.02 0.10 0.47 0.06
0.15 2,100 1,200
0.06 0.05 600
0.10 0.04 0.05
1,300 0.47 0.47
1,700
0.13
0.12
0.47
0.02
0.16
500
1,700
0.17
0.16
0.10
1,800
It will be noted that the solution calls for destination D 6 to be totally supplied from
source Ri. Now, if in fact the capacity of the channel from Ri to D 6 were limited to,
say, 1,000 units, that fact could be reflected by constructing the original problem in
the manner shown in Table H. Here the destination D 6 has been artificially split into
354
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two synthetic destinations, one of which can be supplied from Ri and the other of
which cannot be so supplied. The symbol "M" simply means a prohibitively large
cost associated with the sourcedestination combination, and might have been carried,
for example, as 9.99. Individual variable limitations, whether of fact or of policy,
might be indicated in this fashion.
SupplyDemand Imbalance. Other costs may also be readily entered into the model.
Thus assume that in the previous problem to the given transportation costs are added
manufacturing costs of:
10.0 at Ri 10.2 at R 3
9.9 at R 2 11.1 at R 4
Then the total cost problem becomes as stated in Table I. This looks to be quite
different. In the original problem, the transportation costs were such that the order
of preference of sources for individual destinations was mixed, but with R 4 being first
choice in all cases. In the combined cost problem, R 4 is least preferable, and the costs
have been so chosen that in all cases the order of preference is Ri over R 2 , R 2 over R 3 ,
and R 3 over R 4 . Yet this is the same problem and has the same solution. Reflection
will show this to be true in general, since in any situation of balanced supply and
demand the transportation costs are the only quantities subject to optimization, for
the total output of all plants is required, regardless of the cost level involved.
In a situation of unbalanced supply and demand, however, the model based on
combined manufacture and transportation costs may give an optimal solution which is
quite different from that based on transportation costs alone. Then the model tech
nique may be used to assign the proper level of production at each of the plants, when
not all the production of all plants is required. It is especially interesting to note that,
when the "incremental" production at different plants is at quite different price levels,
it is not necessarily true that the lowestcost production is to be used in the optimal
solution.
Assume, for example, that in the previous illustration an imbalance of supply and
demand is created by expanding demand by 10% at all destinations, and expanding
supply by 20%. Further assume that the additional, "incremental" supply is available
at a cost which is 80% of the cost of the basic supply at the same plant. The result
ing distribution problem is represented by the schedule shown in Table J.
To represent the imbalance of supply and demand, a fictitious destination, D n , can
be introduced, with a fictitious demand equal to the difference between total supply
and true demand. At the same time, the difference between production costs of the
basic and incremental productions can be reflected by the difference in the cost entries
for Ri and for the increments for Ri, R 2 , R 3 , and R 4 . Finally, the entry "M," having
its previous meaning of a very large cost, associated with the fictitious destination for
basic production and the zero entry for fictitious demand and incremental production,
can be used to ensure that all basic production is employed before the incremental
production, dependent on the basic production, enters the solution.
One solution to this problem (there are numerous solutions with equal costs, al
though none with lower cost) is shown in Table K. The numbers in bold face rep
resent costs to the destination.
Unexpected Findings. Surprising things may transpire in investigating optimum
costs with a model. For instance, under the particular cost conditions assumed in the
preceding illustration, all the additional production required came from the least
costly incremental production available. But such action is not always desirable. To
demonstrate this, let us vary incremental unit costs only slightly, while keeping them
in the same relative order. Let unit production costs of the additional 20% be as
follows :
8.00 at Rx (Base capacity cost 10.0)
7.95 at R 2 (Base capacity cost 9.9)
8.10 at R 3 (Base capacity cost 10.2)
8.55 at Ri (Base capacity cost 11.1)
The problem thus becomes that of optimizing the possibilities shown in Table L.
356
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MATHEMATICAL MODELS IN CAPITAL BUDGETING 361
One solution (again there are numerous alternates) is shown in Table M.
Note that this solution calls for using fully the incremental output of R, although
this is most expensive, while using none of the incremental output of R 2 , which is
seemingly the cheapest. Such a result may seem contrary to "common sense," but
only because in the "commonsense approach" costs are separately defined and ex
amined, without respect to the entire system.
This result, unexpected as it is, makes one wonder if it might not be desirable to
reduce the "base" production at one plant while expanding the "incremental" capacity
at another. This possibility can be examined in the model by discarding the concept
of incremental capacity, recognizing only used and unused capacities, and assigning a
cost to each. The cost of production becomes a fixed quantity, equal to the average
unit cost at full production, regardless of the level of production actually used. The
unit cost of unused capacity is also fixed and can be shown to be equal to:
(Unit cost of "base" capacity — unit cost at total capacity) X ("base"
capacity) f ("incremental" capacity)
It will then be noted that unused capacity appears in the solution with a cost as
sociated. Conceptually, this is important and intuitively satisfying. Failure to use an
available facility, the employment of which will increase efficiency, is a neglect of
opportunity which should be penalized. Under this concept, the costs of the various
plants become those shown in Table N.
TABLE N. COSTS OF USED AND UNUSED CAPACITY
Source Ri Rz R s Rj,
Unit cost, base capacity 10.0 9.9 10.2 11.1
Base capacity 6,500 2,000 4,800 4,800
Unit cost, additional capacity 8.0 7.95 8.10 8.55
Additional capacity 1,300 400 960 960
Average cost 9.666 9.575 9.860 10.675
Unit cost, unused capacity 1.666 1.625 1.750 2.125
And the problem becomes that of optimizing production and distribution in the
system. Table O shows the laid down cost per unit to destination.
Table P shows that not only should the cheap "incremental" production of R 2 not
be used, but the base production should also be cut back, as should the base produc
tion of R 3 , while the apparently expensive base and incremental productions of R 4
should be fully utilized.
The simple model and optimization technique are thus seen to be capable of formu
lating reasonably complicated managerial questions — and of demonstrating rather
surprising answers.
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A MODEL FOR SCALE OF OPERATIONS 363
XX.
A Model for Scale of Operations*
Edward H. Bowman and John B. Stewart]
IMPLICATIONS OF SCALE OF OPERATIONS
The president of a wellestablished New England company recently asked
one of those infrequent but important questions — "Should we add some
warehouses to our existing distribution system?" The answer proved to be
a dual one: a no to the specific problem he had in mind; and a yes to one
not suspected. Fortunately, the method of solution employed helped pro
vide answers for both problems. Anticipated savings from the first problem
alone were 10 percent of net annual profits.
The usefulness of this method is significant in two respects. First, com
panies with similar problems may find some of the ideas directly applicable
to their situations. Second, the general method employed is illustrative of a
way of thinking that has broad potential application to business problems.
Some similar methods have been receiving increasing management interest
under the name of "operations research."
For instance, the same general procedure could be used to determine the
best size for salesmen's territories, the best number of branch production
facilities, or other issues which are fundamentally problems in the scale of
operations.
THE PROBLEM
OPTIMUM WAREHOUSE TERRITORY
This New England company had acquired more than a dozen warehouses
in the five states served from its manufacturing plant. Warehouses had been
added in some areas and discontinued in others as changing conditions
seemed to dictate. These decisions had been made on a "common sense
basis," and often had proved advantageous to warehouse operations in one
area but at too great a cost in reduced operating efficiency in adjoining
areas. The problem was further complicated by the fact that two basic
* From the Journal of Marketing, national quarterly publication of the American
Marketing Association, Vol. 20, No. 3, 242247. Reprinted by permission of the
Journal of Marketing.
t The authors are indebted to The Harvey N. Shycon Company, Management Con
sultants of Boston, for the opportunity to work on this and other interesting problems.
364 FACILITIES PLANNING
delivery methods are used. One type of distribution involves semitrailer
delivery from plant to warehouse, unload and storage, and then individual
deliveries to market, while in the manufacturing plant area, individual or
direct delivery to market is the rule.
The general manager recently raised the question of what was the
optimum size of a warehouse territory. Some executives believed drivers
making deliveries directly from the plant warehouse were spending too
much time getting to and from cities located about an hour's drive from the
plant.
The problem as restated by the analysts was, "How large a territory
should be served by a warehouse to result in a minimum total cost for ware
housing, trucking between plant and warehouses, and delivery from ware
house to customers?" It was imperative that this question be answered from
an overall standpoint — covering the company's complete physical distri
bution system — rather than on an individual territory basis.
CRITERIA FOR OPTIMUM TERRITORY
The real and useful objective of the whole project first had to be deter
mined. Profits would be a good place to start. However, sales for this par
ticular problem were fixed, so to maximize profits was equivalent in this
case to seeking minimum costs. The "measure of effectiveness" chosen,
after examination of warehouse operations, was cost (within the warehouse
district) per dollar's worth of goods distributed. It was this cost which
should be minimized. Initially, minimum cost per warehouse sounds good.
However, this would yield a legion of very small warehouses which when
totaled would give an inefficient total operation.
DETERMINANTS OF COST
Available data were obtained from the company's records. Examination
of these data revealed that the cost of material handled in each warehouse
district appeared to be primarily dependent upon two opposing factors : the
volume of business passing through the warehouse and the area served by
the warehouse. The greater the volume handled, the smaller would be the
cost per dollar's worth of goods distributed. However, the greater the area
served, the greater would be the cost per dollar's worth of goods distributed.
Finding this relationship was, of course, no surprise. The crucial job was
to establish the precise relationship between these factors; that is, their
relative importance and the rate at which their variation affected the over
all economy of the system. To be satisfactory, the analysis would have to
handle both factors simultaneously. This done, it would be possible to
A MODEL FOR SCALE OF OPERATIONS 365
predict the cost of distributing goods as the area served by and the volume
handled in each warehouse changed. More importantly, the systems could
be so arranged that total cost would be minimized.
The analysts recognized that many other variables in this situation could
affect the measure of effectiveness. For instance, the price paid for gasoline
in each warehouse area would affect the cost of operations in the area and
undoubtedly varies throughout the New England states. The particular
design of the warehouse, for example, whether the loading platform was at
tailgate level of trucks or on the ground, might also affect these costs. How
ever, it was desirable to keep the analysis fairly simple and so only the two
factors considered most important were included.
THE WAREHOUSE DISTRICT MODEL
OBJECTIVES AND LIMITATIONS
Because of the great complexity of the "real world," models of aspects of
situations are often built as aids to analysis and understanding. These
models may be physical replicas, electrical analogues, blue prints, charts
and graphs, and so forth. The most abstract and probably the most uni
versal type of model is the mathematical one. It has been widely used in the
physical sciences, in some of the social sciences, and in engineering. It is
well to keep in mind that none of these models are the real world. They
are at best useful simplified representations or abstractions of it. Mathe
matical models were chosen for this particular real world because of the
precision with which they could portray the relationships involved and with
which they could reveal minimumcost solutions.
ELEMENTS OF THE WAREHOUSE
DISTRICT MODEL
To build the mathematical model, it was necessary to understand the
economics of the problem. Warehousing costs per dollar of goods handled
tend to decrease with increasing volume: costs of supervision and other
overhead are spread over more units, labor can usually be used with a
lower proportion of idle time, etc. Since distance traveled would be the
main factor determining costs associated with area, it followed that this
cost would tend to vary approximately with the square root of the area.
(Radius and diameter vary with the square root of the area of a circle.)
As concentric rings of equal area are added, rings rapidly become narrower,
that is, additional distance traveled becomes smaller.
366 FACILITIES PLANNING
MATHEMATICAL EXPRESSION
To summarize, it had been determined that for the problem at issue here,
the cost per dollar's worth of goods distributed (the warehouse efficiency)
was equal to certain costs which vary inversely with the volume plus certain
costs which vary directly with the square root of the area plus certain costs
which were affected by neither of these variables. Putting this last factor
first, these same variables arranged as a mathematical expression are as
follows :
(1) C = a{^+cVA
Notation: C = cost (within the warehouse district) per dollar's worth of
goods distributed — the measure of effectiveness.
V = volume of goods in dollars handled by the warehouse per
unit of time.
A = area in square miles served by the warehouse.
a = cost per dollar's worth of goods distributed independent of
either the warehouse's volume handled or area served.
b = "fixed" costs for the warehouse per unit of time, which
divided by the volume will yield the appropriate cost per
dollar's worth distributed.
c = the cost of the distribution which varies with the square
root of the area; that is, costs associated with miles covered
within the warehouse district such as gasoline, truck re
pairs, driver hours, etc.
METHOD FOR SOLUTION
The company had over a dozen warehouses and it was possible to deter
mine for each the cost per dollar's worth of goods distributed (C), the
volume of goods handled by the warehouse (V), and the area served by
the warehouse (A ) . Then, by the statistical method of leastsquares multiple
regression, it was possible to use this warehousing experience and to deter
mine mathematically the values of the coefficients or parameters a, b, and c,
which will make the model the closest predictor of the actual cost for all
present warehouses using the individual volume and area figures. 1
1 This method minimizes the sum of the squares of the differences between the
actual cost and the predicted cost. An expression for this is as follows:
Minimize:
Ci ( a+ ^r +c ^
iN
\ 1
where C, Vi, and Ai indicate actual values in a given (the ith) branch warehouse
operation and the 2 indicates a sum total of all (N) warehouses. Actually a set of
three simultaneous equations are solved for a, b, and c.
A MODEL FOR SCALE OF OPERATIONS
367
HOW GOOD WAS THE WAREHOUSE
DISTRICT MODEL?
In order to confirm the accuracy of this model, a Cost (C) was com
puted for each warehouse using the determined values of a, b, and c from
the multiple regression calculation and the warehouse's specific figures for
V and A. By comparing these computed costs with the actual warehousing
costs, the correlation coefficient was found to be .89, indicating a fairly high
degree of correlation.
MINIMIZATION WITH THE WAREHOUSE
DISTRICT MODEL
CONVERSION OF THE MODEL
Developing this model is only the first step making it possible to predict
costs. What is desired is cost minimization. However, in this case, it was
necessary to convert a part of the model mathematically in order to minimize
it. The object was to express cost as a function of only one unknown
(area). A relationship was found between volume and area for each section
of New England. This sales density (K), expressed in dollar volume per
square mile of area, is
(2)
K =
Therefore, V = KA, and it is possible to substitute this expression for V
in the original model, giving
(3)
^ KA
N
Cif a +
K b . c ^)
3c
N
1
3a
N
Cl (a+A + c y— )] 2
=
=
These equations establish what the statistician calls the normal equations.
368 FACILITIES PLANNING
where a, b, and c are now specific figures determined from the multiple
regression calculation.
COST MINIMIZATION WITH RESPECT TO AREA
What is desired now is an expression for A which will make this cost
model a minimum. It is possible to do this by differentiation which gives 2
/2/)\2/3
(4) A
=(f) ;
This expression for the area A indicates that area which would yield a
minimum cost and is a function of b and c (costs calculated from the
empirical data) and K (the sales density of the area in question).
The cost which is minimized is the explicit cost within the warehouse
district. The implicit cost, interest on investment in inventory and equip
ment, was analyzed and demonstrated to be insignificant for the purposes
of this study. The costs also did not include the cost of loading semitrailer
trucks at the plant and transporting them to the branch warehouses, since —
as long as goods are handled from a branch warehouse — these costs will be
incurred and will not be affected by volume handled or area served by each
branch warehouse.
BRANCH WAREHOUSE FINDINGS
The company's actual branch warehouse areas ranged from about 95 to
150 percent of the individually computed optimum areas. This disclosed an
answer to a question not originally framed, namely, that most of the branch
warehouse areas were too large and that, therefore, there were not enough
warehouses in outlying districts.
THE PLANT DISTRICT MODEL
COST RELATIONSHIPS
The problem of the area to be served from the plant warehouse was
distinct from the branch warehouse problem since goods which are dis
tributed directly from the plant do not have to be loaded into semitrailer
trucks and transported to the branch warehouses and then, unloaded, super
vised, and stored in that warehouse. Therefore, as this area served from the
plant is increased, these costs are saved. However, increasing this area
makes it more expensive to serve the increasingly distant perimeter areas
The expression for differentiation is
dC _ be
dA kA 2 2 \/A
A MODEL FOR SCALE OF OPERATIONS 369
from the plant because more of the delivery driver's time and the truck's
miles are spent in unproductive time driving to and from the delivery area.
MARGINAL MODEL
The type of model set up for this problem was a marginal model. The
plant warehouse area should be expanded out to the point where the cost
of serving the marginal area (the last addition) from the plant is equal to
the cost of serving it from an optimally placed branch warehouse. Reducing
the plant warehouse area from this line would mean that intervening cus
tomers served by a branch could be served more cheaply from the plant.
Expanding the plant warehouse area beyond these points of marginal
equality would mean that the additional customers then served from the
plant could be served more cheaply from branch warehouses.
ELEMENTS OF THE MARGINAL MODEL
This model of marginal equality, the cost of handling one piece of goods
for the marginal area, follows:
Plant direct delivery cost per piece = branch delivery cost per piece (that is,
plant to branch plus branch to customer) .
ITp Pd +T f + T d H d _ S t + B e + 2S D P + S f + 2S d H m D p + /,,
P„(H d 2P d H m F t ) P s
2T D h + T f + T d H d
+
P h (H d 2D, ) H lll F t )
Notation: T„ = Truck operation cost per mile
P,i — Plant delivery miles
T f — Truck fixed costs per day (amortization type charge)
T d — Truck driver costs per hour
F t — Fixed driver time per day (check in, check out, coffee break,
etc.)
D,, — Miles from branch to delivery
Si = Semi load and unload costs
B e = Branch expense per semi
S„ — Semi operating costs per mile
D p = Miles from plant to the branch
S t — Semi fixed costs per day (amortization type charge)
S d — Semi driver costs per hour
I w = Inventory costs per semi per week
H d = Hours per day
H m = Hours per mile
P h = Pieces per hour
P s = Pieces per semi
370 FACILITIES PLANNING
Both sides of the equation ultimately resolve to expressions of cost per
piece. The distance D b to the marginal district from the optimally placed
branch warehouse is determined from the general model, that is:
From this expression for optimum area may be computed an approximation
of the optimum radius (A = n r 2 ) which is the figure used in the model
for D h . The fixed branch expense per semi, B ( , was determined from the
value for b in the general model. The figure 2 in each case represents a
round trip.
PLANT DELIVERY FINDINGS
Most of the cost expressions in the plant district model could be deter
mined directly from the company's internal records. Several of the costs
such as a truck fixed cost per day or cost per mile were checked also from
outside sources. After all the specific values were inserted in the model, it
could be solved for P df the distance from the plant to the farthest district
within the plant warehouse area. 3
Solving the model above for the problem posed indicates that the
marginal boundary of the plant warehouse area should be extended out to
approximately two and one fourth times the present area radius and that
the area to be served from the plant warehouse is thereby increased to five
times the original size. Thus, the initial question is answered with an em
phatic no; that, far from decreasing the plant delivery radius, the company
can best be served by increasing that radius.
SUMMARY
GENERALIZATION OF THE METHOD
As many business problems will yield to similar methods of analysis, the
particular approach found useful here may be generalized as a sequence of
steps as follows:
( 1 ) Following a study to determine the economics of the problem, a measure
of effectiveness was selected.
(2) A mathematical model of the problem was built around this measure of
effectiveness and included those variables which most appeared to influence
the measure of effectiveness.
3 The equation was an implicit expression for P,i. This resulted in a quadratic form,
which was then solved for P,i.
A MODEL FOR SCALE OF OPERATIONS 371
(3) The coefficients in the model were chosen by mathematical manipulation
(multiple regression) to make it as accurate a symbolic description as
possible.
(4) The model was "tested" by statistical means (correlation coefficient).
(5) Again by a mathematical manipulation (differentiation), the model was
minimized with respect to the factor to be used in the decision rule. That
is, an expression for area was determined, which should give the minimum
warehousing cost per dollar's worth of goods distributed in each particular
area.
(6) In the special case of the plant warehouse area, an equation of marginal
analysis was employed to establish the optimum radius to be served directly
from the plant.
VARIATIONS WITHIN THE METHOD
Some variations within the same general framework of analysis might be
considered:
(a) The measure of effectiveness might have been cost per pound of goods
distributed or some other similar criterion.
(b) The model might have included more variables such as warehouse design,
etc.
(c) The coefficients in the model might have been determined from an engi
neering or cost accounting type of approach. From a chart of accounts and
past records and budgets, a, b, c, etc., might have been determined. This
type of approach is more common in business today. However, by defini
tion (the statistician's), these values could have been no better and might
have been poorer.
(d) A tabular or graphical comparison of costs "predicted" by the model to
actual warehouse costs might have been used rather than the correlation
coefficient.
(e) A tabular, graphical, or trial and error method might have been used to
determine the A (area) which minimized the cost expression. However,
this would vary with K — the sales density — and, therefore, it would have
been necessary to repeat this procedure for selected values of K.
(f) Rather than set up an equation for marginal analysis in the special case of
the plant warehouse, it would again have been possible to tabulate, graph,
or try many radius distances with their associated costs to determine the
best one.
IMPLICATIONS OF THE METHOD
The length of time necessary and the accuracy of results for these differ
ent methods of analysis would probably vary with the person using them.
It is suspected that the more conventional approach would have been sub
stantially more time consuming and probably less precise. Possibly the best
advice is: if the more elegant shoe fits, wear it.
Appendix A: An Introduction to
Linear Programming
Linear programming is a reasoning device which systematically evaluates
plans, determines whether improvement is possible, and if so, suggests
changes to increase the value of the original plan. Thus, linear program
ming lays a path leading toward optimization where optimization is defined
by a maximal or minimal value. Although originally conceived by applied
mathematicians, it is essentially a trialanderror reasoning device. Its use,
therefore, is not restricted to mathematicians. Linear programming is
limited only in the ability to express relationships in linear mathematical
terms and to conceive problems in a programmed fashion.
Specifically, linear programming provides a method to determine the
best combination of a series of unknown variables where the solution must
lie within certain stated bounds. The bounds (or constraints) may be
financial (budgetary), or may involve time (equipment capacity), or
quantity (amount shipped). The most straightforward example of linear
programming occurs in determining how much of what products to produce
when production is limited (constrained) by capacity. For example, con
sider the Nix Company which produces a deluxe and a standard model
Vostok. 1 The deluxe model contributes $3.00 per unit to overhead and
profit, and the standard model, one dollar.
The standard model requires 2 fabricating hours per unit to produce,
occupies 4 square feet per unit in the warehouse, and does not require
any painting. The same figures for the deluxe model are: 3 hours, 3
square feet, and 1 hour to paint. Available fabrication time is limited to
no more than 24,000 hours; warehousing space is limited to no more than
36,000 square feet; and time in the painting shop is limited to no more
than 6,000 hours. An optimum solution requires the determination of the
1 This example is based on a problem from R. Stansbury Stockton, An Introduction
to Linear Programming, Allyn and Bacon, 1960.
372
AN INTRODUCTION TO LINEAR PROGRAMMING 373
number of standard and/or deluxe models to produce in order to maximize
the total contribution to overhead without violating the fabricating, ware
housing, and painting constraints.
In the Nix Company problem, the constraints are in terms of fabricating
hours, warehouse space, and painting hours. Let S stand for the number
of standard models to be produced and D for the number of deluxe:
Fabrication hours: 25 + 3D ^ 24,000 (A.l )
Warehouse space: AS + 3D ^ 36,000 (A.2)
Painting: ID ^ 6,000 (A.3)
(A.l) says (in quantitative terms), 2 hours per standard unit times
the number of standard units produced plus 3 hours per deluxe unit times
the number of deluxe units produced must be less than (or at most equal
to) the 24,000 hours of fabricating capacity available. (A.2) says that the
amount of warehouse space used to store the standard model (4 square
feet per unit times the number of units) plus the space required for the
deluxe model (3 square feet times the number of deluxe units produced)
cannot exceed the 36,000 square feet available. Finally, since the standard
model does not require any painting time, the number of deluxe models
times 1 hour cannot exceed the 6,000 painting hours available — (A.3).
The problem is to maximize the total contribution which can also be
expressed as a linear function. Each standard unit produced contributes
$1; while each deluxe model contributes $3. Any solution, therefore, which
maximizes Z = IS + 3D and is subject to the constraints, will provide an
optimal solution. Z is the total contribution consisting of $1 times the
number of standard models produced plus $3 times the number of deluxe
models produced. (Z is indefinite because S and D are as yet undetermined.)
The problem can be expressed concisely in mathematical terms as
follows :
Maximize: IS f 3D = Z
Subject to: 25 + 3D ^ 24,000
45* + 3D ^ 36,000
lDf^ 6,000 (A.4)
One additional constraint must be explicitly stated. The amount of either
S or D produced must be positive or zero. Negative production (mean
ing that units are deproduced or dismantled) is nonsensical. This con
straint is stated as follows:
S ^ and D ^
or S,D^0 (A.5)
A graphical model provides the easiest way to see the solution to this
problem. Consider a two dimensional graph measured in units of S and
374
APPENDIX A
units of D. The fabricating constraint appears as line (1) in Figure Al.
Line (1) is located as follows:
24 000
If D = 0, that is, if no deluxe models are produced, S = —
= 12,000 standard units. In other words, if all of the 24,000 available
fabricating hours are allocated to the standard model, then 12,000 standard
models can be produced because the standard model requires 2 fabricating
hours per unit. By the same reasoning, if S = 0, that is, if no standard
24 000
units are produced, D = —  = 8,000. The D intercept of 8,000 units
and the S intercept of 12,000 units are connected with a straight line be
cause the constraint is linear. Similarly, line (2) is positioned in Figure
Al with intercepts of 9,000 for S (D = 0; and S = 36 ' 00Q ) and 12,000
for D (S — 0; D
36,000
)•
Finally, the painting constraint does not affect the amount of the
standard model produced, because zero hours of painting capacity are
required to produce the standard model. Hence, the maximum amount of
12000
10000
\ (2) Warehouse
8000
Q
UNITS OF
o
o
//y^z/K,
(3) Printing
p °S/y
4000
A
+
B
R V<^/
2000
K^
w (1) Fabricating
V
^x
2000 4000 6000 8000 10000 12000
UNITS OF S
FIGURE Al. NIX CO.
AN INTRODUCTION TO LINEAR PROGRAMMING 375
the deluxe model which can be produced is limited to no more than 6,000
units (6,000 hours divided by 1 hour per deluxe unit). In linear program
ming language, the painting constraint is "not binding" on production of
the standard model.
The solution to the Nix Company problem lies in the area bounded by
the vertical axis, the horizontal axis and the shaded area in Figure Al.
A point in this area has coordinates which represent values of S and D.
For instance, point A represents D = 4,000 and S = 2,000. Point B
represents D = 3,000 and S = 4,000. Since a value of less than zero is
precluded by ( A.5 ) , S ^ 0, the solution cannot lie to the left of the
vertical axis where S would be negative. Similarly, any point below the
horizontal axis would have a negative value for D and would be prohibited
by constraint (A.5).
Any point beyond the innermost set of lines would likewise violate
constraint (A.4). For instance, any point beyond line (3), D ^ 6,000,
would call for more production of D than the paint shop could provide.
Therefore, any point outside the polygon 0PQRS is not feasible because
of the capacity and nonnegativity constraints. The search for an optimal
program can, therefore, be limited to points lying within or on the polygon
0PQRS.
Actually, an even stronger statement can be made. The optimal solu
tion, which maximizes the contribution function IS f 3D = Z must lie
on the polygon PQRS. Any point closer to the origin than the perimeter of
PQRS does not provide for the maximum feasible production and, there
fore, represents a less than optimal plan. As an example, a point between
and P on the vertical axis cannot be an optimum program since more
deluxe units could be produced without violating any constraints. These
extra units would yield additional contribution and thereby would in
crease Z.
The same analysis holds for any point lying between and S. There
fore, the optimum program must lie on PQRS because a program lying
within 0PQRS will not yield the maximum amount and, hence, will not
optimize Z. Because PQRS prescribes the location of a feasible program, it
is called a feasibility polygon.
Let us solve for the values of S and D at point R where line (1) in
tersects line (2). We can obtain values for S and D at this point by solving
the equations for lines (1) and (2).
2S + 3D = 24,000
4S + 3D = 36,000 (A.6)
The applicable inequalities in (A.4) have been changed to equalities
in (A.6). As discussed previously, any optimum feasible solution will lie
376 APPENDIX A
on the polygon PQRS; or in other words, any optimal solution will utilize
all of the available production capacity. If all of the capacity is utilized,
the equation which represents the programming of that capacity will not
leave any excess capacity (as indicated by the "less than" sign), but will
call for the elimination of any excess, usable capacity. The total pro
grammed usage, therefore, will equal that available, and the ^ in
(A. 4) becomes an = in (A. 6).
The solution can be obtained by multiplying the first equation in (A. 6)
by —2 and adding to the second:
45 + 3D = 36,000
_4S 6D = 48,000
_3D = 12,000
D = 4,000
Substituting D = 4,000 in (A.4),
45 + 3(4,000) =36,000
45 = 36,000  12,000
45 = 24,000
S = 6,000
Therefore, at point R in Figure Al, D = 4,000 and 5 = 6,000 units.
The Z value at point R is $18,000, determined as follows:
6,000 of 5 @ $lea. = $ 6,000
4,000 of D @ $3 ea. = 12,000
Z = $18,000
The student should verify that at point Q, the total contribution is $21,000;
at point P, $18,000; and at point S, $9,000. These values can be deter
mined either graphically or algebraically as above.
Thus, point Q represents the optimal program because the Z value at that
point is $21,000. The Z value or contribution function can be shown as a
line passing through point Q. To completely describe the line, another
point, preferably an intercept on one of the axes, is needed. This point
can be determined as follows:
At point Q the value of Z is $21,000. The point on the vertical axis
which is also on the Z — $21,000 line is found by substituting a value of
5 = (indicating an 5 = point on the vertical axis) into the contribu
tion function.
Z = 21,000 = 3D + 15 = 3D + 1 (0)
D = 21,00 ° = 7,000 deluxe units
AN INTRODUCTION TO LINEAR PROGRAMMING 377
The horizontal intercept of the Z line can be determined in a similar
fashion:
Z = 21,000 = 3(0) + 1(S)
Z — 21,000 standard units
This line is called an isoprofit line for Z = $21,000. Any point on this
line, representing a combination of S and D, will yield a contribution of
$21,000. Similar lines can be drawn for any other value of Z, all of which
will be parallel to the original line. If the slope of the contribution line is
exactly equal to the slope of any line forming the feasibility polygon,
PQRS, then a range of optimum programs can exist. For instance, if the
slope of the contribution line for Z = $21,000 is exactly equal to the slope
of line QR, then any point on QR represents an optimum feasible solution
to the problem.
This analysis essentially completes the logic of linear programming. Un
fortunately, very few linear programming problems can be expressed in
only two dimensions as in the Nix Company problem. Therefore, let us
move directly from the two dimensional problem involving two products
to the n dimensional problem where there are any finite number of products.
THE SIMPLEX METHOD
The simplex method can be used to solve any linear programming prob
lem. Unfortunately, the simplex method involves some unsophisticated,
but complex, mathematics; therefore, we must now digress briefly to
examine the mathematics behind the general linear programming problem.
In general, the linear programming problem exists because there are
more unknowns than equations. If we had one equation per unknown, the
unknowns could be determined by solving the equations simultaneously. But
in the general linear programming problem, there are m equations (m is
a positive, finite number) and n unknowns. Furthermore n > m (n is
greater than m); that is, there are more unknowns than equations.
We can operate on such an equation system by invoking the following
theorem:
If there exist any nonnegative solutions to a system of m equations (with n
unknowns) then at most m unknowns in the solution are positive and the rest
are equal to zero. 2
For instance, consider the following equation system:
2X 1 + 4X 2 + 3Z 3 + 5X 4 = 26
X t + X 2 + 2X 3 + 4X 4 = 10 (A.7)
2 A rigorous proof of this theorem stated in somewhat different terms can be found
in Dorfman, Samuelson and Solow, Linear Programming and Economic Analysis,
McGrawHill, 1958, p. 75.
378 APPENDIX A
In this equation system, there are four unknowns, X l9 X 2 , X 3 , and Z 4 ,
and only two equations. According to our theorem, only two of the four
variables can be positive, and the rest must be equal to zero. Assume
that X x and X 2 are positive and, therefore, that X s and X A are zero. Simply
removing both X 3 and X 4 from (A. 7) :
2X t + 4X 2 = 26
X ± + X 2 = 10 (A.8)
Solving these equations simultaneously by subtracting 2 times the second
from the first,
2X 1 + 4Z 2 = 26
2X 1 — 2X 2 = 20
2Z 2 = 6
Therefore, X 2 = 3. Substituting X 2 — 3 in (A.8)
2^ + 4(3) =26
2Z X =2612
2X 1 = 14
Zx =7
(It makes no difference into which equation of (A.8) X 2 = 3 is sub
stituted.) In other words, if X 3 and Z 4 equal zero, then X x = 7 and
Z 2 = 3. A solution to (A. 7) is, therefore,
X 4 =
* 3 =
Z 2 = 3
X 1 = 7
The variables Z x and X 2 which are «ttf sef equal to zero are termed basic
variables. Together they comprise a basis for equation system (A.7).
We can use this basis to evaluate the Z value for (A.7). Assume
AX 1 + 2Z 2 + 5X 3 + X 4 = Z
Therefore,
Z = 4(7) +2(3) +5(0) +0
Z = 28 + 6
Z = 34
The selection of X 1 and X 2 as a basis was somewhat arbitrary. Now we
know that the X lt X 2 basis yields a Z value of 34. If we wished, we could
try a basis of X x and X s and compare its Z value to 34. Similarly, we
could evaluate bases of X x and X 4 , X 2 and X 3 , X 2 and Z 4 , and finally
X% and Z 4 . In this equation system the number of combinations of pairs
AN INTRODUCTION TO LINEAR PROGRAMMING 379
of variables is fairly limited; however, in a system of five equations with
fifty unknowns, there are 2,118,760 possible bases! Thus we require a
systematic procedure to select the basis.
Let us refer to the original equation system.
4X 1 + 2* 2 + 5* 3 + X, = Z
2X 1 + 4* 2 + 3* 3 + 5*4 = 26 (A.8.1)
X 1 + X 2 + 2*3 + 4*4 = 10 (A.8.2)
Step 1: Subtract (A.7.2) from (A.7.1) to create a coefficient of 1 for * x
in (A.7.1). Preserve all equations intact except (A.7.1).
4* x + 2* 2 + 5*3 + *4 = Z
*! + 3* 2 + * 3 + * 4 = 16 (A.9.1)
^ + X 2 + 2*3 + 4*4 = 10 (A.9.2)
Step 2: Subtract (A.9.1) from (A.9.2) and subtract 4 times (A.9.2)
from the Z equation to create a coefficient of zero for *i in all
equations except (A.9.1). Preserve (A.9.1) intact.
—2*2  3*3 — 15*4 = Z  40
X x + 3* 2 + * 3 + * 4 = 16 (A.10.1)
2*2 + * 3 + 3*4 = 6 (A.10.2)
Step 3: Multiply (A.10.2) by — 1/ 2 .
—2* 2 — 3*3  15*4 = Z  40
*! + 3*2 + 1*3 + * 4 = 16 (A.ll.l)
* 2  */ 2 Xs  3 / 2 ^4 = 3 ( A. 1 1 .2 )
Step 4: Add 2 times (A. 11.2) to the Z equation. Subtract 3 times
(A.11.2) from (A.ll.l).
_4^ 3 _ 18*4 = Z — 34
^i + %^2 +5y 2 *4 = 7
* 2 + 2i/ 2 * 3 +5y 2 *4 = 3 (A.12)
By virtue of our theorem, * 3 = * 4 = 0. Therefore, the revised
equation system is
0=:Z34
* 1 = 7
* 2 = 3 (A.13)
Note that the solution, *i = 7 and * 2 = 3 appears explicitly in the
equation system. Moreover, since
= Z'— 34
Z = 34
380
APPENDIX A
The system in the form of (A. 12) is a canonical or reduced form of
the original system. The coefficients of the canonical form are relative to
the coefficients of the original form. Moreover, the solutions of the ca
nonical system are identical to the solutions of the original system. This
useful property has already been demonstrated. The variables which ap
pear with a coefficient of f 1 in the canonical system are the basic variables.
Let us apply this concept to a soap scheduling problem:
A soap factory makes four detergents, Multi, Certo, Fluffi and Permi.
It has four process centers, 101, 201, 301, and 401. The time in hours
per ton required for each product in each center is :
M
101
201
301
401
The production planning department informs you that the available
capacity in the process centers is 6,000 hours in 101, 9,600 in 201, 8,800
in 301, and 4,400 in 401. Multi sells for $12 per ton, Certo for $8,
Fluffi for $6, and Permi for $2 per ton. How much of what products
should be produced?
The problem is framed as follows. The variables X x . . . Z 4 , (that is
Xx through X 4 ) are called slack variables. They take up the slack in
each process center (row). Thus, the "less than or equal to" in the
original problem becomes an "equal to."
M
C
F P
x 1
X 2
* 3
x,
5
4
10
1
1
= 6,000
. =9,600
8
2
1
= 8,800
4
1
1
= 4,400
For example, the constraint in the case of process center 101 is:
5M+ IOP.^6000
AN INTRODUCTION TO LINEAR PROGRAMMING 381
whereas in the format above,
5M + 10? + lXt = 6000
Variable X 1 represents any time on process 101 which is not utilized by
the programmed amount of M and P. The same kind of analysis holds
for 201, 301 and 401. Note that the dimension (units) of X t . . . X 4 are
hours of time on the respective process center. In contrast, M, C, F, and
P, the real products, are programmed in tons.
The slack variables comprise a basis, because their coefficients are
1 in one equation and zero in all others. Thus, a feasible program is:
X = 6,000 hours
X = 9,600 hours
X = 8,800 hours
X = 4,400 hours
Clearly, we can improve profitwise on this solution by programming a real
product, that is, by bringing in a real product to utilize (replace) the slack
product (unused capacity).
Step 1: Which product should be produced?
Real product M is worth $12 per ton. Let us, therefore, bring
M into the program.
Step 2: How much of M can be programmed?
The amount of M which can be programmed is limited by centers
301 and 401 (rows 3 and 4) which can produce no more than
1100 tons. Therefore, row 4 governs the amount of M to bring in.
Let us bring M into the basis by creating a coefficient of 1 for
M in row 4, and zero for M in rows 1 through 3. Refer to the
original equation system.
X2 X3 X4
6000 (A.14.1)
1 9600 (A.14.2)
1 8800 (A.14.3)
1 4400 (A.14.4)
Step 2 A: Multiply (A.14.4) by 2 and subtract from (A.14.3).
M C F P X x X 2 Z 3 Z 4
5 10 1 6000 (A.15.1)
4 1 9600 (A.15.2)
1 2 (A.15.3)
4 1 1 4400 (A.15.4)
M
C
F
P
x 1
5
4
10
1
8
2
4
1
382
APPENDIX A
Step 2B: Multiply (A.15.4) by % and subtract from (A.15.1)
M
F
5
4
1
P
10
x 1
1
X,
Step 2C: Divide (A.16.4) through by 4
M C
F
P x x
5
4
10 1
4
1
y 4
The basis is
now
x u x 2 ,
from (A.17)
is
x<
X,
x,
5
4
500
(A.16.1)
9600
(A.16.2)
2
(A.16.3)
1
4400
(A.16.4)
*4
5
4
500
9600
2
y 4
1100
(A.17)
X n and M. M replaced X A . The solution read
IX t . = 500 /ic?«r5
1A^ 2 = 9600 /zowrs
1Z 3 = Ziowrs"
1M = 1100 roiu
All other variables, according to our theorem, are equal to zero. Notice
that the dimensions are hours for the basic slack variables and tons for
the real basic variable.
Step 3: What product should be brought into the program at this stage?
The answer to this question is product C which is worth $8 per
ton. But in order to establish a general procedure, we are going
to enlarge the (A.17) system.
Cr>
12
8
6
2
L'H'
Basis
M
C
F
P
*i
x 2
*3
x,
x t
5
4
10
1
5
4
500
x 2
4
1
9600
X 3
1
2
12
M
1
%
%
1100
The "basis" column is simply a listing of the variables in the basis at the
AN INTRODUCTION TO LINEAR PROGRAMMING
present time. The "C" row is a listing of the revenue coefficients, in dollars
per ton in this case. These C/s are the coefficients in the Z equation. Since
slack time contributes no profit, the C/s for columns 5, 6, 7, and 8 are zero.
Letting / take on a subscript to denote the column, C 5 = C 6 = C 7 = Cg = 0.
Similarly, d = 12, C 2 = 8, C 3 = 6 and C 4 = 2.
The "C" column lists the C } for the basic variables. Thus, the d in row
1 is the Cj for variable X x which equals zero. Letting the subscript i take
on the row number, C 2 = 0, C 3 = 0, and C 4 = 12 because the coefficient
of product M in the Z equation is 12.
We now define a Z as follows : Zj is the d for a row times the coefficient
for that row within the tableau, summed by column. Thus, the Z value in
the F column (Z 3 where / = 3), is equal to
Column
d
F
(times)
5
4
=
(times)
=
(times)
=
12 (times)
y 4
= 3
4 columr
z 3
= ?
Similarly Z 8 (the Z value for the X
is equal to
Column
d
X,
(times)
5
~~ 4
=
(times)
=
(times)
2
=
12 (times)
Ya
= 3
z 8
=7
Finally, a C, — Z ; row is defined as follows: Q — Zj equals the Z value
/or the column (Zj) subtracted from the C value at the head of the column
(Cj). Thus C 3 — Z 3 (the F or third column) equals 6 — 3 = 3, and
C 8 — Z 8 (the eighth or X 4 column) equals — 3 = — 3. The C, — Zj
value represents the marginal revenue for that column.
Let us examine the Cj and Z } values in more detail. First consider the
values in (A. 18). Each X value represents the marginal rate of substitution
of the column variable for the row variable. For example, the technical
384 APPENDIX A
constraints state that producing one ton of M will use up 5 hours of X\.
That is, each unit of M programmed at the stage where X x = 6000 hours
requires 5 hours. Similarly, each unit of M requires 8 hours of X 3 and 4
hours of X±.
c,
12
8
6
2
Basis
M
C
F
P
X,
x 2
X*
x,
x 1
5
10
1
6000 hrs.
x 2
4
1
9600 hrs.
X s
8
2
1
8800 hrs.
x,
4
1
1 4400 hrs.
(A.18)
The marginal rate of substitution of C (column 2) for X 2 (row 2) is 4.
That is, it takes 4 hours of X 2 to produce one ton of C. The reader can also
verify that the marginal rate of substitution of F for X 8 is 2 and of F for
Z 4 is 1. The marginal rate of substitution of P for X\ is 10.
We defined the Zj value as the product of d times the row X, summed
by column. Z) for column 1 (product M) equals
Basis
c x
M
Xi
(times)
5
X 3
(times)
8
x,
(times)
4
Zi =
=
Since the values in the M column represent the marginal rates of substi
tution of M for X l9 X 2 , Xz, and X 4 , the Z value for the M column repre
sents the revenue given up for one unit of M produced. The tableau shows
that 5 hours of X lf 8 hours of Z 3 , and 4 hours of X 4 are given up to pro
duce one ton of M. Since X u X 3 and X 4 bring in zero dollars per hour, no
revenue is given up to produce one ton of M.
The Cj represents the revenue per ton for the column variable. For
example, the revenue per ton for M is $12; for C, $8; for F, $6; and for
P, $2. If, in the case of M, a marginal ton can be produced for zero dollars
given up (the Z, value), then the marginal revenue for one unit of M is
$12 — $0, or $12. In general terms, the marginal revenue is Cj — Z, (once
again, because C, equals the revenue per ton brought in and Z, equals the
revenue given up to produce one unit of the column variable). Referring to
tableau (A.18), the marginal revenue for the M column is zero dollars
because it is already in the basis. The marginal revenue for the C column is
$8; for the F column, $3; and for the P column, $2. The marginal revenue
AN INTRODUCTION TO LINEAR PROGRAMMING 385
for the Z 4 column is $ — 3, representing a marginal cost of $3. (The Zj
value represents the sum of the revenues given up to produce one ton of the
column variable or the foregone revenue for one ton of the column variable.
The Cj value represents the revenue per ton for the column variable. There
fore, the Cj — Zj value represents the revenue per ton minus the foregone
revenue per ton or the marginal revenue per ton of the column variable.
Thus, if Cj — Zj is negative, the revenue given up {Zj) is greater than the
revenue obtained (Cj), and Cj — Zj is a marginal cost.) The reader may
find it necessary at this stage to reread and study tableaus (A. 14) through
(A. 18) in order to master the reasoning associated with the simplex
method.
As augmented, the tableau representing a firsttrial solution is
Cj
12
8
6
2
Q
Basis
M
C
F
F
x 1
x 2
X s
x 4
x 1
x 2
4
5
4
10
1
1
5
4
500
9600
X s
1
2
M
1
Vi
%
1100
Zj
12
3
3
CjZj
8
3
2
3
(A.19)
The marginal revenue for product C (C 2 — Z 2 where / = 2) is $8.
Therefore, C is the next product to bring into the program. No more than
2400 tons (9600 divided by 4) can be produced. Product C is brought in
by creating a zero in rows one, three and four of column two and by making
the 4 in row two a 1. Fortunately, the zeros are already there. Therefore,
we need only divide row two by 4.
Cj 12 8 6 2
d Basis M C F P X t X 2 X 8 X±
8
C
1
5
4
10
1
y 4
500 hours
4
2400 tons
Xs
1
—2 hours
12
M
1
Va
/4
% 1100 tons
Zj
12
8
3
2
3
Cj — Zj
3
2
2
3
(A.20)
386 APPENDIX A
Thus, X 1 = 500 hours
C = 2400 tons
X s = hours
M = 1100 tons
The marginal revenue for F is $3. (See the "3" in C 3 — Z 3 .) Since C 4 — Z 4
(product P) equals $2, product F should be brought into the program.
Multiply row four by 5 and add to row one to create a zero in row one,
column three.
c,
12
8
6
2
Ci Basis
M
C
F
P
*i
x 2
^3
x 4
X 1
5
10
1
6000 hours
C
1
Va
2400 tons
Xs
1
—2 hours
M
1
y 4
i/ 4 1100 tons
Multiply row
four by
4.
C,
12
8
6
2
Ci Basis
M
C
F
P
x 1
*2
x*
X,
X t
5
10
1
6000 hours
8 C
1
%
2400 tons
X s
1
—2 hours
6 F
4
1
1 4400 tons
Zi
24
8
6
2
6
Cm 7
12
2
2
6
Notice that F has replaced M in the previous basis. That is, a real product
replaced a real product.
What product next? Product F should be brought in because it has a
positive marginal revenue of $2. (See C 4 — Z 4 = 2.) Divide row one by
10 to bring P into the basis. No other reduction is necessary.
Q
12
Ci
Basis
M
2
P
y 2
8
C
*3
6
F
4
Zi
25
C —7 
13
6 2
F P A^i ^L2 ^3 ^4
1
10
600 tons
y 4 2400 tons
1—2 hours
1 1 4400 tons
6 2 — 2 6
10
2
10
2
AN INTRODUCTION TO LINEAR PROGRAMMING
387
This tableau presents an optimal program because all Cj — Zj values
are negative. No product can increase the present Z value because none has
a positive marginal revenue. The basis at this stage is composed of P, C,
X 3 and F.
IP — 600 tons
\C— 2400 tons
1Z 3 = hours
1/7 _ 4400 tons
In summary, let us describe the simplex method in a flow diagram form:
1. State the problem as a system of linear equations and a linear func
tional to be maximized or minimized.
2. Determine a basic feasible solution (usually consists of the slack
variables).
3. Determine Zj, the marginal foregone revenue, for each column.
4. Subtract each column Z } from its coefficient in the linear functional, Cj.
IF MAXIMIZING
IF MINIMIZING
5a. Is there a (Cj — Zj)
which is positive
(greater than zero)?
5b. Is there a (Cj — Zj)
which is negative
(less than zero)?
■V ir V V
6a. If no, STOP. 6b. If yes, select 6c. If no, STOP. 6d. If yes, select
(Basic feasi
ble solution
is optimal
program.)
most positive
Cj — Zj and
bring that
variable into
the basic solu
tion.
(Basic feasi
ble solution is
optimal pro
gram.)
most negative
(CjZj)
and bring that
variable into
the basic
solution.
7a. Recycle to block 3 above.
Why bother with the simplex method? First, it is more methodical, espe
7b. Recycle to
block 3 above.
388 APPENDIX A
cially in the exchange steps, than the initial method. Second, the simplex
method is the general method in use at the present time. Third, the simplex
method displays more information. For example, as noted previously, the
Cj — Zj value is equal to the marginal revenue for the product represented
by that column. Thus, the cost of producing product M is $13 per ton
because its marginal revenue is $—13 (the equivalent of a marginal cost).
Similarly, the marginal cost of X x is $.20 because its marginal revenue is
$— 2 / 10 . Since X x represents slack time on process center 101 (row 1), the
marginal cost of not having one hour on process center 101 is $0.20. In
other words, the opportunity cost of process center 101 is $.20 per hour.
Similarly, the opportunity cost of center 201 (product X 2 ) is $2 per hour,
and the opportunity cost of center 401 is $6 per hour. Thus the simplex
method provides much more than the basic program information.
GENERALIZING THE LINEAR PROGRAMMING
PROBLEM
Thus far the discussion has centered around the solution of specific,
relatively elementary problems. Much of the existing literature in linear
programming consists of discussions of methods of solution and short cut
techniques for the general LP problem rather than a specific problem. We
can round out the discussion of the Simplex Method by introducing a
general problem and the accompanying notation enabling us to think of
linear programming in terms other than the conditions of a specific example.
The general problem may be stated as follows: maximize (or minimize)
the function:
Z zzz C\X\ j C 2 X 2 j . . . f C n X n
Z is a dependent variable which is a function of (depends upon) the inde
pendent variables x x . . . x n (read x x through x n ). The independent variables
are subject to constraints :
fluJfi + a l2 x 2 + • •
#21*1 + #22*2 + • •
. + a ln x n ^ b x
• + a 2n x n ^ b 2
Qm\X\ f a m2 x 2 \ • •
X\, Xo, . . . , X n
• + a mn x n ^ b
In this notation system, the first subscript refers to the row of the term
and the second subscript, to the column. Thus, a 12 is the a for row one,
column two. a mn is the a for the m th row, n th column. Consequently,
a mn x n is the product of a for the m th row, n th column times x for the n tn
column. The Z value is, therefore, equal to c 1 x 1 \ c 2 x 2 + . . . + c n x n where
the dots refer to columns between column two and column n, the last
column.
AN INTRODUCTION TO LINEAR PROGRAMMING
389
In short form, the subscripts i and / are used to refer to a column and a
row. Thus OijXj (/ = 1, / = 2) means ai 2 x 2 or the /4Z product for the first
row, second column. By use of this short form notation, the general linear
programming problem can be expressed as follows:
Optimize :
Subject to:
£j — C IjXj
anXj *= bi
= 1,.
. . , m)
Xj^O
0=1..
.,«)
(A.21)
The (i = 1, . . . , m) (/ = \, . . . , n) notation means "with i running from
1 through m and / running from 1 through n."
Referring to (A.21) and following the general Simplex rules, we can
indicate the general method of solution as follows:
Convert the m constraint inequalities to m equations by inserting m slack
variables, x n+1 . . . x n+m
011*1 + #12*2 + • • • X n + i = b\
#21*! f" #22* 2 \~ • • • X n + 2 = t>2
or in tableau form:
X\ X2 . . . x n
an a 12 ... a ln
(121 ^22 • • • #2n
a m\ a m o . . • Omn
Xn + m — " r
*n+l X n+ 2
X n + m
1
Recalling the important theorem on which the Simplex Method is based,
we know that the optimal solution of the general problem will involve no
more than m nonzero unknowns (or *'s). Stated in another manner, we
know that the optimal solution will involve at least n unknowns with a value
of zero. At this point it must not be forgotten that there are a total of
n \ m unknowns — n original unknowns plus the m slack variables, one for
each constraint equation.
The Simplex Method of solution consists of choosing an initial feasible
solution with m nonzero unknowns (as mentioned previously the typical
first solution is to consider the slack variables as nonzero and all real
variables, x 1 . . . x n , as zero valued) and proceeding in stepwise fashion to
test the existing solution for optimality, moving to a new feasible solution
with an improved Z when the solution is not optimal. By this method, an
optimal solution will always be reached in a finite, although sometimes very
large, number of steps.
The two example problems we have discussed have both been maximizing
problems in which we have been attempting to maximize some profit or
390
APPENDIX A
contribution function. Minimizing problems, to minimize a cost function
for example, can also be solved by the Simplex Method. The technique for
minimizing an objective function (the Z function) is very similar to that
used for maximizing. In fact, the techniques developed for problems re
quiring Z to be maximized can be used directly for minimizing problems by
simply maximizing — Z (which is equivalent to minimizing Z) and re
writing the constraints so that the inequality sign is in the same direction as
in maximizing problems. For example:
rewrite Minimize : Z = Ci^i + ^2*2 "+"■•• + c «*»
Subject to: a n ii + a 12 x 2 . . . a ln x n ^ b x
as: Maximize: — Z = — c^i — c 2 x 2 — ... — c n x n
Subject to: — anx x — a 12 x 2 — ... — a ln x n ^ bt
In tableau form, the minimizing problem can be written:
Cj » — d
— c 2 .
• —C n
.
.
x t
x 2 .
. x n
Xn + 1
X n + 2 •
. . x n + m
—an
— «12 • •
• — «ln
1
h
— 021
— 022 •
. — ct 2n
1
b 2
■a m2
1
b m
Occasionally, linear programming problems will be formulated so that
they contain constraints which are redundant, or worse, not compatible.
Two examples can be shown graphically. Suppose two constraints of a
certain problem are:
(i)
(2)
2x x ^= x 2
*1 — *2 + 1
Constraint ( 1 ) says the solution must be above line ( 1 ) . It is clear, how
ever, that if constraint (1) is satisfied, constraint (2) will also be satisfied
since a point cannot be above line (1) but below line (2). Constraint (2)
in this case is redundant.
An example of conflicting constraints would be:
(1) x,^x 2 + 1 x
(2) Xl ^x 2 + 2
AN INTRODUCTION TO LINEAR PROGRAMMING 391
The first constraint says the solution must be above line (1), and the
second, that the solution must be below line (2). Clearly no point can
satisfy both constraints. In this case the constraints conflict. Fortunately,
most management problems properly formulated will not result in such
situations.
The Dual Problem
Every linear programming problem has an associated dual problem which
may be viewed somewhat as a "mirror image." If the original linear pro
gramming problem, called the primal, is a maximizing problem, its dual
will be a minimizing problem. The constant coefficients in the objective (Z)
function of the primal become the constant column of the constraint equa
tions of the dual and vice versa. In addition, rows of coefficients in the
constraint equations become columns of coefficients in the constraint equa
tions of the dual.
If the primal problem is one of maximizing a profit function where
equipment capacity is constrained, then the dual problem will be one of
minimizing the value or "shadow price" which should be assigned to each
unit of process capacity. For example, the dual might yield a result saying
that the minimum value of one hour of machine A's time is $6. If there is
a job which can be done on machine A which will produce at least $6 of
profit per hour on the machine, it can be scheduled on machine A. If, how
ever, a job which would produce only $5 profit per hour on machine is
the best available, it should not be scheduled.
An excellent discussion of the dual problem can be found in:
R. Dorfman, P. Samuelson, and R. Solow
Linear Programming and Economic Analysis
McGrawHill Book Co., Inc., 1958.
A. Charnes, W. Cooper
Management Models and Industrial Applications of
Linear Programming, Vol. I
John Wiley & Sons, Inc., 1961.
A good example of a primal and dual problem similar to the problem
discussed in this appendix can be found in:
H. Bierman, L. Fouraker, and R. Jaedicke
Quantitative Analysis for Business Decisions
Richard D. Irwin, Inc.,
Homewood, Illinois, 1961.
Appendix B: The Fundamentals
of Calculus
Calculus is a basic branch of mathematics which is extremely useful in
solving maximizing and minimizing problems. It determines the rate of
change of a dependent variable with respect to an independent variable.
For example, calculus can tell us how total cost changes with respect to
the volume of goods produced. Calculus can also be used to determine the
maximum or minimum value of a dependent variable, for example, cost,
with respect to an independent variable, such as the production rate. This
discussion will not attempt to develop proficiency in calculus; it will simply
describe necessary calculus concepts so that the reader can appreciate the
application of calculus to production problems.
THE CONCEPT OF A FUNCTION
The first fundamental of calculus is the concept of dependency. The
phrase "is a function of" simply means "depends upon." For instance, we
can say "y depends upon x" and connote exactly the same meaning as
"y is a function of x" Thus, the mathematical terminology "is a function
of" is the same as the layman's terminology "depends upon." The mathe
matical notation used to denote "y is a function of x" is y = /(*). which
reads "y equals / of x."
By convention, x is assigned to the independent variable and y to the
392
THE FUNDAMENTALS OF CALCULUS
393
dependent variable. That is, the value of y is a function of (depends upon)
a given value of x. Assume that
y = /(*)
and f(x) = $x — x 2
so that y = Sx — x 2
Given a value for x, the independent variable, we can determine y as the
dependent variable. If x = 1, for example
y = 8x — x 2
y=r8(l)(l) 2
y = S— 1
y = 7
If * is 2,
y = 8* — x 2
y = 8(2) (2)2
y= 164
y=12
A table of values for * and y is given below:
X
1
2
3
4
5
6
7
y
7
12
15
16
15
12
7
We can graph these values of x and y as follows:
Y
394 APPENDIX B
In Figure Bl, the dimension on the horizontal axis equals the value of
x and the vertical axis the value of v. Thus, any point on the curve can be
located by specifying its x and y values. Any pair of such values belongs to
the function if the point representing the coordinates of x and y lies on the
curve. For example, the coordinate x — 4, y = 16 — denoted (4,16) (the
x value by convention is specified first) — lie on the curve, but the point
(4,18) does not. This statement can be verified by examining Figure Bl
and also by substitution in the original function. Substituting x = 4 in the
function yields:
y = Sx — x 2
y=8(4)(4) 2
y = 32 16
y= 16
Thus, the point (4,16) belongs to the function. However, the point (4,18),
when substituted in the function, yields:
y z=z Sx — x 2
y=8(4) (4) 2
18 = 32 16
18^16
(^ means "does not equal")
Therefore, (4,18) does not belong to the function. Any point can be tested
in the same way.
In addition to having the properties described above, functions also
exhibit a slope. The slope is the rate of change of the dependent variable,
y, with respect to the rate of change of the independent variable, x. By
convention, the rate of change is called the slope when the change in x
equals one unit.
Let us determine the rate of change between the points (1,7) and
(4,16). Between those points, y increases by 9 units, that is from 7 to 16,
and x increases by 3 units, from 1 to 4. Let us denote the increase by a
A sign (to signify change). Thus Ax = 3 and Ay = 9. The slope is, there
Av 9
fore, = — = 3 units of y for each unit of x. According to this measure,
Ax 3
y changes three times as fast as x, that is, at the rate of three to one.
Consider the rate of change between the points (1,7) and (2,12). In
Ay 5
this interval Ax == 2 — 1 = 1 and Ay = 12 — 7 = 5. The slope — = »
Ax 1
that is, y changes five times as fast as x. To get still another measure, con
sider the rate of change between (1,7) and (3,15). In this interval,
Ax = 3 — 1 = 2 and Ay =15 — 7 = 8. Therefore, the slope equals 4.
THE FUNDAMENTALS OF CALCULUS 395
Ay
Clearly the slope as determined by — depends upon the interval on the
A*
function over which the measurements are made. For summary purposes,
these results are presented in tabular results below. Clearly, we could form
Point
(U)
Ay Ay
as many — ratios as we cared to select intervals. The — ratio is, therefore,
Ajc Ajc
not an exact measure of either the rate of change of the function or the
slope.
Let us narrow our attention to the immediate neighborhood of the point
(1,7). By reducing the Ax interval, we also reduce Ay, and obtain a more
exact measure of the slope at point (1,7). Observe the following table.
Ay
Point
A*
Ay
Ajc
(2,12)
1
5
5
(3,15)
2
8
4
(4,16)
3
9
3
(5,15)
4
8
2
(6,12)
5
5
1
Ay
Point (y — Sx —
x 2 )
Ax
Ay
Ax
(1.5,9.75)
.5
2.75
5.50
(1.4,9.24)
.4
2.24
5.60
(1.3,8.71)
.3
1.71
5.70
(1.2,8.16)
.2
1.16
5.80
(1.1,7.59)
.1
.59
5.90
Point
(1,7)
Apparently we have still not selected a small enough value for Ax to get an
Ay
exact measurement of the slope, because — is still increasing when
Ajc
Ay
x = 1.1 and Ax = . 1 . Furthermore, as we reduce Ax, the slope, — , con
Ax
tinues to increase. Clearly some other tack is required to obtain the exact
and unique value of the slope at (1,7).
Let (x , y ) denote the point at which we desire to determine the exact
slope. Denote the slope by the letter b.
Ax
y = y — yo = f(x) — f(x )
But /(*) = f(x + Ax)
Therefore, Ay = f(x + Ax) — f(x )
396 APPENDIX B
u Ay f ( Xo + Ax> > ~ ' ( * o) rvt 1 ^
And b = — = (B.l)
Ax Ax
In the case where /(x) = 8x — x 2 ,
f(x + Ax) = 8(x + Ax)  (x + A*) 2
=
8x + 8Ax — x c
2 2XyAX 
(Ax)
2
Substituting in (B.l)
b.
8a:
+ 8Ax
— x 2 — 2x Ax — (Ax) 2
Ax
8x
Ax
*o 2
Breaking the
complex fraction into
a series
of fractions
b =
8*0
Ax
8Ax
Ax
Ax
2x Ajc
Ax
(Ax) 2
Ax
8x
Ax
Xo 2
Ax
Finally, dividing as appropriate by Ax, and cancelling
b = 8 — 2x — Ax
The slope, b, at point x is dependent upon x and Ax, but not on y or Ay.
Thus,
Ay
b — — = 8 — 2x — Ax (B.2)
Ax
The slope, therefore, depends upon the interval Ax which we more or less
arbitrarily chose in the previous examples. The act of choosing a Ax, how
ever small, introduced the approximation which prevented us from reach
ing a final decision about the slope.
Let us, therefore, use the measure of slope obtained in (B.2), but simply
let Ax = 0. The slope b would, therefore, be measured exactly at point
(*o, Jo) without introducing the inaccuracy of selecting an arbitrary Ax.
Under those conditions, the slope equals
b = 8 — 2x — Ax
= 82x
= 82x
At the point where x = 1, the slope b equals 8 — 2(1) =8 — 2 = 6.
We were getting close to the exact measure of slope in our previous example
Ay
where Ax = . 1 and — =5.90 and was increasing. Since the slope is
Ax
dependent only upon the value of x , we can determine b at any point on
THE FUNDAMENTALS OF CALCULUS 397
the function.
x b — 8 — 2x
1 6
2 4
3 2
4
5 2
6 4
We notice that the slope decreases to zero and then becomes negative.
Moreover, from Figure Bl, we can observe that the slope is zero where
the curve is a maximum — at point (4,16). We will use this concept shortly.
Let us review briefly the ideas which have been suggested. The entire
analysis was based on the concept of a function. A function indicates
dependence — conventionally y depends upon x or y = f(x). The slope of
Ay
the curve representing this function is denoted b and is equal to — . Since
Ax
Ay = y — y = f(x) — f(x ),
L fix)  f(x )
b =
Ax
If y = f(x) — Sx — x 2 (and y„ = 8x„ — x Q 2 ),
b = 8 — 2*o — Ax
Precisely at point x Q , Ax = 0, and
b = 8 — 2x () .
The slope at point * (where Ax = 0) is called the derivative of the func
Ay
tion. Technically, the derivative is the rate of change, — , as A* approaches
AX
zero. However, for our purposes, we can define the derivative as the slope
at x where Ax — 0.
The derivative is usually denoted as follows:
1 dy
1. — , or
ax
2. — [/(*)] , or
dx
3. f(x)
398 APPENDIX B
Thus, if y = f(x) = Sx — x 2 ,
dy
1. — = 82*
<2Jt
2. — [8jc  a: 2 ] = 8  2x
ax
3. f (*)  g  2*
Fortunately we need not work out the derivative for each function we
face. Someone else has cataloged derivatives determined just as we deter
mined — [8* —x 2 ] = 8 — 2x in (B.l) and (B.2). First, we can catalog
dx
four general rules (a is a constant; g(x) is also a function of x).
1. ^[fl/(*)] = fl^ [/(*)]
dx dx
2. y [/(*) + *(*)] = — [/(*)] + — [*(*)]
ax ax ax
a* dx
d
r [/«]
dx
We can also catalog some specific rules.
4. If /(*)=«, —/(x)=0
5. H/(i)=*, — /(*) = 1
ax
6. If /(*)=*•, —fix)=a^^~
dx dx
6'. If /(*) = jc ffi , — f(x) = o^ 1
Let us use these rules in some examples.
1. f{x) = ax 2 + bx + c
By rule 2,
d d d d
— f(x) = — ax 2 + — £* + — c
dx dx dx dx
THE FUNDAMENTALS OF CALCULUS 399
By rule 1,
d d d d
 r f(x)=a r x* + b r x + c :r l
dx dx dx dx
By rule 6',
d d d
— f(x) =za2x+ b — x\c— 1
dx dx dx
By rule 5,
d d
— f(x) =a2x + b + c—l
dx dx
By rule 4,
d
— f(x) = a2x + b +
dx
Therefore,
d
— [ax 2 + bx + c] = lax + b
dx
2. A more meaningful example —
A = average (unit) cost
q = quantity produced
100 q
A = + —
q 625
100
Since — lOOg 1
Q
By rule 2,
dA d d ( q \
d<? dq dq \625 /
By rule 6',
<L4 d_ / q
■ MX»,:r ' : 4   )
dq dq \625/
By rule 5,
£L4 1
=ioo, 2 + 
400
Therefore,
APPENDIX B
dA
= 100^ 2 +
625
Many different derivatives can be determined with the simple set of rules
given here.
FINDING MAXIMA AND MINIMA IN
FUNCTIONS OF ONE VARIABLE
Figure Bl indicates that the curve reaches a maximum value for y, the
dependent variable, when x — 4. The table of values indicates the same
conclusion, although the exact point is not determined in either the table or
the graph. We can certainly observe that on either side of point (4,16)
the y value is less than 16, but we cannot be certain at what point the curve
reaches a maximum for y.
Let us turn to the slope concept of calculus to determine the exact point
where y is a maximum. If we adopt the convention of algebra, that values
to the right of zero are positive and to the left, negative, we can apply a
sign to the slope value. In Figure B2 1 line OA has a negative slope, and
line OB has a positive slope. Line OC has a larger slope than either OA or
OB; in fact, OC has an infinite slope. The slope of both ON and OP is
1 The concept of Figure B2 was first suggested to the authors in a set of class
notes by W. Starbuck of Purdue University, Krannert Graduate School of Industrial
Administration.
THE FUNDAMENTALS OF CALCULUS
401
zero — the sign makes no difference. Beginning with ON, the slope increases
to a maximum at OC and decreases to zero at OP. Thus, OA has a larger
slope than ON; and OC than OA, but the slope of OB is smaller than OC.
Bringing this concept to the function y — Sx — x 2 , we observe in Figure
B3 that the slope decreases to zero at the maximum point and then goes
negative. In other words, the slope of the curve is positive to the left of the
Slope zero
negative
Figure B  3
point where the curve changes direction and then becomes negative. One
can induce that at the maximum point, called an inflection point, the slope
is zero at the instant when it goes from + to — . In this function, the
inflection point provides a maximum value for y.
But the derivative also gives the slope of the curve. In this case, when
y = %x — X 2 ,
dx
= 82*
At the inflection point, the slope equals zero; so we simply set the derivative
equal to zero:
dy
— = 8  2x =
dx
and solve for the value of x.
S = 2x
4 = x
Plugging x = 4 into the functional expression, we obtain a value for y.
y=&x — X*
= 8(4)
= 16
(4)
402 APPENDIX B
Therefore, the calculus indicates that the values of x = 4, y = 16 (point
4,16) is a maximum value for y. (Since we obtained a value for y from the
expression y = Sx — x 2 , we are guaranteed that the point (4,16) lies on
the function.)
To generalize, one can always obtain the value of the variables at any
point of inflection by setting the derivative equal to zero. However, the
point of inflection could be a maximum or a minimum. If a function has a
maximum and a minimum, setting the derivative equal to zero will yield
two pairs of values for x and y. In general, the use of calculus will yield as
many pairs of values for x and y as there are maxima and/or minima. Those
which are minima can be determined by substituting values for x to either
side of the inflection point and thereby determine the direction in which the
slope is changing. 2
Let us apply this method to find the minimum of
100 q
A=z f —
q 625
where A = average (unit) cost, and q = quantity produced
Earlier we determined that
dA 1
— = 100q 2 +
dq 625
Setting the derivative equal to zero, we obtain
1
lOOtf" 2 =
625
100 1
^ 2__ 625
q 2 = 62,500
q = V62,500
q = 252 (approximately)
Substituting q =
252 into the function,
100 252
A = + = .80
252 625
(approximately)
2 Actually, the test is better accomplished by taking the derivative of the derivative,
termed the second derivative, and denoted £j. If ^ <0 at (x , y ), then /(jco) is
a maximum. If _Z > at (jCo> yo)> then /(xo) is a minimum. If ^ = 0, then
f (x ) may be a maximum or a minimum and the test suggested above might be tried.
THE FUNDAMENTALS OF CALCULUS 403
Substituting q = 200 into the function, we get
A = .82 (approximately)
Therefore, since the point (200,. 82) lies above the point (252,. 80), the
derivative found a minimum.
FINDING MAXIMA AND MINIMA IN
FUNCTIONS OF SEVERAL VARIABLES
We have been working with functions of one variable, that is, where y
depends only upon x. In many cases, y depends upon several independent
variables. Assume, for example, that y = f(x, z) (y is a function of x
and z).
y = fix, z) = x s + 5z + 10
The slope of y with respect to x considered solely is the partial derivative
dy
of y with respect to x, denoted — . The rate of change of y with respect
dy
to z is denoted — . The rule for partial derivatives is to consider all
dz
variables but the particular one under consideration as constants. Where
y = fty, Z ) = X s + 5z + 10
dy
consider z a constant when taking — and consider x a constant when
dx
dy
taking — . Since the derivative of a constant is zero,
dz
?>y
— = 3jc 2 + = 3jc 2
dx
Similarly,
^ = + 5 = 5
dz
(The reader should verify these partial derivatives by using the rules for
simple derivatives.)
Partial derivatives are in a sense a special case of simple derivatives.
If there is more than one independent variable, there will generally be one
equation for each derivative (or as many equations as independent vari
ables)
. Examples:
1.
f(x,y, z) —
yx 2
+ ZX + C
dx "
d
dy ~
d
dz '
2yx +
a: 2
z
2.
f(x,y) =
(x
aY{y
2>) 2
dx ""
2(jc —
a)
(y
by
dy "
2(y
b)
'(x
a) 2
APPENDIX B
3. We can also illustrate partial derivatives in a simple inventory model.
Consider an item in inventory with the following properties :
A = the reorder cost for the item including order processing and manu
facturing setup cost
s — the annual rate of usage in units
i = the inventory carrying cost in $ per unit per year
q = the size of the ordered lot
Since the inventory level varies between zero and q, the average inventory
q
level is — . With an inventory carrying cost of i, the annual cost is
q
Annual carrying cost = i— (B.3)
The annual usage is s units and the number ordered per lot is q; therefore,
the number of times per year that the item will t
cost per order is $^4, the annual ordering cost is
the number of times per year that the item will be ordered is — . Since the
Annual ordering cost = A~ (B.4)
q
The total cost, C, is therefore the summation of the ordering cost and the
carrying cost, or (B.3) plus (B.4).
THE FUNDAMENTALS OF CALCULUS 405
Q S
C = i+A (B.5)
2 Q
To minimize the total cost, take the partial derivative of C with respect to
each independent variable. First with respect to i (employing the rule that
the derivative of a product is the sum of the derivatives of each term times
the others)
2? = f (B.6)
di 2
The partial derivative of i with respect to itself is 1. Therefore, since all
other variables are considered constant and the derivative of a constant is
zero, all the terms but one "wash out" and leave only
dC q
— =  for (B.6).
di 2
Taking the partial derivative of C with respect to q in (B.5),
q s
2^ q
dC i
— = Asq~ 2 (B.7)
dq 2
(B.7) is significant because it gives the rate of change of C with respect
to q. To minimize C, set (B.6) equal to zero and solve.
dC i
— =   Asq~ 2 =
dq 2
2
Asq~ 2
i
2 =
As
q 2 =
2As
i
n —
l2As
(B.8)
\ i
The reader would do well to verify (B.6, B.7 and B.8) using the derivative
rules and the concept of a partial derivative. (B.8) is the traditional eco
nomic lot size formula. Using this formula to determine the lot size q will
406 APPENDIX B
minimize the total cost of storing and ordering the item of inventory.
SUMMARY
1 . The derivative is a mathematical expression which gives the exact slope
at a point on the function.
2. Since the slope at either a maximum or minimum is zero, the maxima
and minima can be obtained by setting the derivative equal to zero and
solving for the undetermined variables.
INTEGRATION
Integration is the reverse of differentiation. Whereas in differentiation,
we determine the mathematical expression for the slope of a function, in
integration we determine the function given the slope of the function.
For example, can we infer f(x) if we know that — = 8 — 2x? Let us
dx
try f(x) = Sx — x s
A [gjc _ x 3 ] = 8  3x 2
dx
Therefore, f(x) = Sx — x s is not the function. Next, try f(x) = Sx — x 2
— [8*  x 2 ] = 8  2x
dx
Therefore, it appears that the function f(x) — Sx — x 2 goes with the
derivative 8 — 2x. But the derivative of fx(x) = 20 + 8* — x 2 is the same
as the derivative of f(x) = Sx — x 2
f(x) =Sxx 2 hix) = 20 + Sx  x 2
A [s x _ x 2] = s _ 2x A [20 + 8jc  x 2 ] = 8  2x
dx dx
Thus, the inference process of finding an fix) which yields — fix) = 8 —
dx
2x is not entirely the answer, because the derivative of any constant is 0.
Therefore, the derivative of fix) = A \ Sx f x 2 is the same as the
derivative of fix) = Sx — x 2 where A = any constant.
Sometimes we can solve for A. For example, if we know that the point
(4,16) lies on the function, then A must equal zero.
THE FUNDAMENTALS OF CALCULUS 407
£ [A + 8jc  x 2 ] = 8  2x
dx
16 = ,4 + 8(4)  (4) 2
16 = ^ + 3216
A =
In general, however, in rinding the function by inference from the
derivative, we must always beware that an unknown constant may be part
of the function.
The process of finding the function when the derivative is known is
called integrating. Integration is denoted by
/
( )dx
1. ${2ax + b)dx = ax 2 + bx + A
consequently, — (ax 2 + bx + A) = lax + b)
dx
The A cannot be determined in this case.
2. f(bx~ 2 + cx~ 3 )dx = bx 1 —  cx~ 2 + A
because — (bx 1 cx~ 2 + A) = — 6x 2 + cx~ 3
dx 2
The A cannot be determined.
3. /(100^+^ = 100^ + ^
because — (IOO4 1 + 2 + /4) = lOOtf 2 + —
d# 625 625
Fortunately, we need not go through the inference process each time an
integral is required. Just as there are elaborate derivative tables available,
there are also detailed integral tables. We need only remember that
dA
— = 0, and, therefore, that each integral may have an unknown constant.
Index
Index
[Numbers in italics refer to footnotes in the text]
Ackoff, R. L., 333
Andress, Frank J., 188
Anticipation stocks, 225, 239
Anshen, Melvin, 106
Anthony, Robert N., 309
ArnorT, E. L., 333
Arrow diagramming, 10, 1 1
Backhauling, 64
Base stock system, 22122
(see also Inventory controls)
Basic feasible solution, 387
Basis, 378
Beckman, M., 120, 129
Bennion, Edward G., 329
Bierman, H., 391
Blackwell, David, 328
Blanchard, R., 129
Bliss, Charles A., 53
Bottleneck concept, 15
definition, 1415
examples of, 15, 79, 81, 85, 91
importance, 15
resolution of, 1517
Bottlenecks (supply), 35354
Boulanger, David G., 130
Boulding, Kenneth E., 328
Boundedvariables problem, 1 02
Bounds (see Constraints)
Bowman, Edward H., 102, 310, 363
Breakeven analysis, 178
Breakeven point analysis, 24245,
249, 25056
breakeven point, 244
chart, for three alternatives, 254
combined with economic lotsize
theory, 25056
Brown, F., 129
Bryson, Martin R., 279
Bryton, B., 173
Bullinger, C. E., 256
Burgeson, J. W., 173
Calculus (Appendix B), 392406
dependency concept, 392
function of, 392, 397
derivative, 397
partial, 403
second, 402
integration versus differentiation,
406407
maxima and minima
one variable, 400403
rules, 398
examples of, 398400
slope, 395
411
412
INDEX
Capital budgeting, 3143 1
Capital budgeting problem, 33044
Capital investment programming prob
lem, 33641
Carnegie Institute of Technology, 10,
119
Cell routes, 104
Central limit theorem, 157
Central service bureau, 74
Chamberlain, Clinton J., 152
Characteristics
sales, 23738
production, 23839
Charnes, A. C, 10, 53, 75, 84, 90, 93,
102, 105, 391
Charts
bar, 131
milestone, 131
Chung, Anmin, 10
Churchman, C. W., 333
Clark, C. E., 139
Cohen, Kalman Joseph, 258
Commodity distributiontransportation
problem, 34762
Commonsenss approach, 36163
Concave programming, 77
Condition equations, 30
Constraints, 11, 106, 121, 37274,
38890
capacity, 305
conflicting, 390
critical path, 1 112
redundant, 390
slack paths, 1112
Control charts, 291
illustration, 298
IPS, 293300
method, 29293
procedures, 294
Cooper, W. W., 10, 53, 75, 84, 90, 93,
102, 105, 391
Cost components, 11112, 154, 336,
366
Cost of conversion (see Transporta
tionproblem procedure)
Cost table, 9495, 98, 336'
Costs:
fixed, 24344, 249
minimum, 24849, 25152
total, 244, 247
variable, 24445, 249
Costs for decision making, 8687,
10912, 122
gasoline warehouse problem, 8687
production costs, 334
supplydemand imbalance, 35558
Crane, R., 129
Crash problem (see Shortpeak season
problem)
Critical path method, 142
Critical path scheduling, 11, 144
Cumulative probability function, 275
78
Cupola charge problem, 41
description of, 4143
example of, 43
formulation of mathematics, 41—45
linear programming applied to, 41,
45
long method of solving, 485 1
Cutandtry methods, 48, 55
Dantzig, G. B., 93, 105
Daum, T. E., 173
Decision factors, 378
Decision rules, 11318
employment — , 11314
linear—, 119, 124
production — , 11314
Decision making, 331
Degeneracy, 95, 9899
coping with, 9899
definition of, 98
example of, 99
Demands
actual, 4
cumulative, 5
expected, 5
forecasted, 45
tableau of, 5
maximum expected, 23 1
rates, 251
weekly, 275278
tableau of, 27577
Departures, 35253
DFPA manual, 3336
Distribution problems, 34445
Dorfman, R., 65, 377, 391
Dual, 35 (see also Linear program
ming)
Dudley, J. W., 290
INDEX
413
Economic lot size, 178
Economic lot size theory, 242, 24549
mathematical expression, 24849
tableau of formulae, 246
Eidmann, F. L., 256
Employment decision rule, 11314
Equiprobability strategy, 330
Error rates, 282, 286
estimated, 286
tableau of, 28586
Event
network, 132
arrowconnecting, 132
Exogenous variables, 315
Expected time estimate, 15657
beta distribution, 15657
standard deviation, 15758
Facilities plans, 30513
alternative, 306
capital budgeting approach, 306,
313
payback period, 306307
simple rate of return, 307308
present value analysis, 309
time value of money, 308309
Fairfield, Raymond E., 193
Farr, D., 93
Fazar, Willard, 139
Feasible solution, 35153
Feasibility polygan, 375
Feller, W., 214
Fetter, 310
Fluctuation stocks, 236
Ford, Henry, 159
Forecasts
converted, 19
cumulative, 232, 23637
errors, 22930, 237
requirements (SOBIL), 27071
sales, 23041
tableau of, 23 1
Forecasts, economic, 31427
forecasted probabilities, 32023
indifference probabilities, 318327
most probable, 3 1 7
most probable cycle phase, 326
probability coefficients, 317
true probability, 32223
Fouraker, L., 391
Free market, 32
Freight yardswitching problem, 120
129
backlog costs, 12223
comparison of costs, 1 2526
total costs, 124
Function of capital, 33354, 338
Gaddis, Paul O., 141
Game theory, 32629
Gantt charts, 7
basic types, 7
load chart, 7, 8
planning or progress charts, 6,
79
illustration, 8
weakness, 9
Gantt, Henry L., 141
Gasoline — blending problem, 7477
(see also General procedure)
Gass, S. I., 10,35
General decision problem, 1 10
General mathematical techniques,
11013
General procedure, 7374, 98, 102
gasolineblending problem, 7477
Girshick, M. A., 328
Grant, E. L., 256
Hamlin, Fred, 39
Harrison, Joseph O., Jr., 105
Heinz Co., H. J., 50, 6069
simplest kind of problem, 6064
double problem, 6571
fixed cost, 66
variable cost, 66
Henderson, A., 10, 53, 102, 105, 186,
319, 329, 333
Hetrick, James C, 331
Heuristics, 12
application of, 12, 160, 16472
definition of, 12
Hoehing, W. F., 293
Hoffman, T. R., 173
Holt, C. C, 10, 106, 118, 119, 124,
129,213
Houthakker, H. S., 105
Hurwicz, L., 328
Hydeman, W. R., 274
Input transaction document, 147
Inputs (see Transportationproblem
procedure)
414
INDEX
Integrated petroleum company refin
ing problem, 31924
Inventory, 282, 28490
shutdown, 290
Inventory controls
examples of applications, 180
fixedorder system, 202
management and control of, 182,
189
planning and scheduling, 182, 189,
241
reasons for, 1 80
techniques for, illustrated, 215
basestock system, 221222
economical order quantities, 216
20
fixed reorder cycle, 22021
warehouse orders, 216
Inventory costs, 1 86
capital invested, 189
direct ( outofthepocket ) , 1 879 1
foregone opportunity, 187, 19091
Inventory functions, 183
basic, 183
customer service, 191
for movement, 184
for organization, 18486
anticipation, 185
fluctuation stocks, 185
lot size, 185
safety stock, 200, 22932, 23537
fixed order system, 202204
periodic reordering, 205
Inventory problems, 18182
Inventory replenishment
fixed order system, 202204, 207
208
periodic reordering, 205208
factors affecting, 22021
tableau of, comparing time, 221
Investment alternative, 31629
IPS (see Control charts)
Jackson, J. R., 173
Jaedicke, R., 391
Karush, William, 264, 21 A
Kendall, D. C, 129
Kilbridge, Maurice D., 159
Klass, Phillip J., 139
Knoppel, C. E., 256
Koopman's, T. C, 93, 105
Lathrop, John B., 256
Least cost solution, 45, 66, 67, 85
Lemke, C. E„ 102
Linear decision rules (see Decision
rules)
Linear programming
advantages of, 51, 54, 5761, 64,
78, 81
cupola charge problem, 41
definitions, 40, 372
double problem, 6571 (see also
H. J. Heinz)
elements of, 9
examples of formulation, 4345
forms used in, 49
functions, 9
introduction to, (Appendix A, 372
91)
limitations, 78
logpurchasing problem, 3338
model, 9
Nix Company problem, 37277
graphical model, 37477
PERT network problem, 13438
productmix problem, 2332
simplest problem, 6064 (see also
H. J. Heinz)
tube mill problem, 1322
uses, 2122
Linear programming problem, 37791
dual, 391
general, 38891
short form, 38990
tableau form, 38990
Line balancing (problem), 12, 159
balance delay analysis, 16164
method of, 16472
tableau of, 165, 168, 17071
terminology, 16061
Log purchase problem, 33
dual of, 35
definition, 35
downgrading, 36
linear programming for, 36
Lost sales, 26667
Lot size problem
size)
Lucas, Kenneth C,
Luce, R. D., 330
Luce, Raiffa, 330
(see Optimal lot
264, 274
INDEX
415
McCloskey, J. F., 105
McGuire, C, 120, 129
McKinsey,J.C. C, 328
Machineshop problem, 7883, 91,
102
Magee, J. F„ 105, 177, 179, 200, 225
Malcolm, D. G., 139
Manne, A. S., 78
Mansfield, Edwin, 119, 129
Manual on Quality Control of Mate
rials, 294
Marginal model, 36970
Marginal rate of substitution, 384
Marginal rates
costs, 8788
profits, 8791
Marginal revenue, 38488
Marginal table, 99101
Mathematical model, 347, 36571
Mathematical programming, 53
{see also Linear programming)
advantages of, 5761, 6471, 85
86,92
case examples, 59—85
effect on company policy, 5459
functions of, 56, 89
limitations of, 5758
uses of information (a planning
tool)
capital investments, 91
cost of improvements, 8990
marketing policies, 89
most profitable customers, 89
product cost, 88
Matrix, or matrices, 16, 2732, 317
20, 32325
definition, 16
distribution — , 1619
construction of , 1617
functions, 16
simplex, 16, 18
Maximax strategy, 330
Maximin strategy, 330
Mellon, B., 75, 84, 90
Metzer, R. W., 39, 52
Miller, Robert W., 140
Mitchell, J., 173
Model {see Mathematical model)
Modigliani, Franco, 10, 106, 11819,
124, 129,273
Monte Carlo approach, 264, 272,
27476
Monthend inventories, 232
tableau of, 233
Moody, Leland A., 264, 21 A
Moranec, A. F., 274
Morgenstern, Oscar, 328
Morse, Phillip M., 264
Muth, J. F., 10, 106, 118, 119, 124,
129
Nonlinear programming {see Concave
programming)
Normal distribution, 263
Objective function, 349
Opportunity profit, 15
Optimize, 333
Optimum lot size, 19299
sample problems, 19399
algebraic solution, 196
graphic solution, 195
Optimum solution, 34951
Orensteen, R. B., 256
Outputs {see Transportation problem
procedure)
Paths
critical, 1112, 132, 13637, 142,
144
determining, 158
parallel, 13132
series, 13132
Payback period, 306307
PEP, 11, 130, 151
PERT, 1012, 130, 140
advantages of, 138, 14548
implementation costs, 14851
level of indenture, 148
management's responsibilities to
ward, 155
network integration, 149
objectives, 131
production application, 14950
program analysis, 133
program development, 13133
(for) projectstructural organiza
tions, 14648
requirements, 14243, 15152
slack order report {see Slack)
416
INDEX
PERT (cont.)
timecost relationships, 15254
vehicle armament system, 134
Poison distribution, 26163
Present value analysis, 309
Present value concept, (see Time
value of money)
Primal linear programming problem,
391
Production
combinations of, 30
ideal, 30
modified, 30
negative, 29
Production control, 3, 4
Production control rule
mathematical expression, 237
Production cycling problems, 209
capacity levels, 213
example, 20910
optimum — versus safety stocks, 210
tableau of, 210
production levels, 21012
Production decision rules, 1 1314
Production facilities, 4
Productmix problem (see Gasoline
blending problem)
Production planning
advanced techniques, 234 (see also
Linear programming)
conventional methods, 5
functions, 3
graphical techniques, 23134
use, 3
Production plans (see also Production
planning)
alternative, 5, 23234, 250
optimal, 7, 250
uniform, 234
Production rates, 5, 251
Production requirements (see also De
mands)
plan I, 57
plan II, 57
Production scheduling, 3, 4, 103, 106,
20915,240
example, 214
paint factory problem, 11118
techniques, 240
Production schedules, 4, 179
Profitpreference procedure, 78, 92
93
Program Evaluator and Review Tech
nique (see PERT)
Quality control, 28084
sample, 282
random, 283
size, 283
Rational decisionmaking mechanism,
33343
construction of model, 33343,
34549
restrictions, 34950
mathematical techniques, 34546
optimization procedure, 35153
setup costs, 34647
Rautenstrauch, Walter, 243, 257
Raymond, Fairfield E., 245, 257
Replenishment cycle, 26164, 278
discrete frequency distribution, 262
Restricted market, 32
Restrictions, 349
Return, 23, 2729, 32
actual, 28
ideal, 28
Reul, Ray I., 335
Rinehart, R. F„ 279
Roseboom, J. H., 139
Safetystock problem (see Inventory
functions)
Salveson, M. E., 174
Samuelson, P., 377, 391
Scheduling problems, 58
Schlaifer, Robert, 53, 105, 186, 319,
329, 333
Schwarzbeck, R., 39, 52
Screw machine problem, 8485
Service failure cost, 204
Servo theory, 21213
Shadow price, 2729
Shortpeak season problem, 226
mathematical approach, 22728
newsboy case solution, 22627
Simon, H. A., 10, 106, 118, 119, 124,
129, 213
Simple rate of return, 307308
Simplex method, 103105, 377 (see
also General procedure)
INDEX
417
Simplex method (cont.)
flow diagram form, 38188
soap scheduling problem, 38087
Simulated Optimal Branch Inventory
Levels (see SOBIL)
Slack
negative condition, 143, 158
order report, 1 44
positive, 158
times, 144
Slack variables, 380
Smith, Wayland P., 242
SOBIL system, 26673
graphical method, 270
procedure, 26869
requirements for use, 2707 1
simultaneous curves, 266
tableau of, 268
Solow, R„ 377, 391
Spearman's rank correlation coeffi
cient, 289
SQC, 28590
SQC versus inventory, 28687
frequency of mistakes, 287
Standard deviation, 237
Starbuck, W., 400
Statistical elapsed time, 133, 135
Statistical variance, 133, 135, 143
Steward, John B., 363
Stigler, George F., 328
Stockton, R. Stansbury, 372
Substitution process, 30
Supply center, 280
Taylor, Frederick W., 141
Terborgh, George, 257
Thusen, H. G., 257
Time estimates
earliest expected time, 135
expected time, 136, 156
latest time, 135
midrange, 15657
most likely, 13235, 143, 15657
optimistic, 13235, 143, 15657
pessimistic, 13235, 143, 15657
probability of meeting, 13637
schedule time, 136
slack time, 13536
Time value of money, 308
formula for calculating, 308
tableau of, 309
Timing problem, 324
Tonge, F. M., 174
Transportationproblem procedure, 73.
92101, 103106
raw material problem, 7274
short method of solving, 93101
advantages, 104105
characteristics, formal, 99101
cost of conversion, 99101
inputs, 99101, 103
outputs, 99101, 104
when to use, 99101
Tree of capital, 340
Trefethen, F. N., 105
True cost, 57, 86, 88, 1 1 1
Variables, 34049, 36566
basic, 37883
slack, 380
Varnum, Edward C, 257
Vassean, H. J., 207, 213
Vazsonyi, Andrew, 264, 214
Villers, Raymond, 257
Von Neumann, John, 328
Waitingline theory, 21415
Warehouse district model, 365
cost minimization, 36768
marginal model, 36970
mathematical expression, 36667
Warehouse efficiency, 366
Warehouse network problem, 25974
optimal inventory level, 268
Warehouse territory problem, 363
Wein, Harold H., 119, 127
Weinstock, Jack K., 264, 274
Wester, Leon, 159
Whitin, Thomson M., 247, 257
Williams, J. D., 327
Winston, C, 120, 129
Woods, B. M., 257
Work force employment plans, 305
Young, David M., 214
ROBERT H. BOCK, Assistant Professor in Produc
tion at Northwestern University, is a graduate of Purdue
University, receiving his B.A., M.A., and Ph.D. from
that institution, with majors in Mechanical Engineering,
Industrial Engineering and Industrial Management, re
spectively. Professor Bock attended the Ford Foundation
Summer Program on the application of mathematics to
business held at the University of Michigan.
WILLIAM K. HOLSTEIN earned his B.A. degree at
Rensselaer Polytechnic Institute, majoring in Chemical
Engineering. He received his M.A. degree from Purdue
University in Industrial Management. He is presently on
the staff at Purdue in the Graduate School of Industrial
Administration and will receive his Ph.D. degree in
Industrial Economics. Professor Holstein has been ac
tive in research, in addition to his assignment as Grad
uate Admissions Coordinator for the Graduate School
of Industrial Administration.
This book is printed in Times Roman type, a highly
legible and versatile masculine face, simple in design
and medium in weight. It was created in 1932 by
Stanley Morison, typographic consultant, for The Times
of London. The text body is complemented by the use
of Bodoni Bold in the part and chapter openings and
reading titles, making a fine contrast in lights and darks
in the total book design.
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MARSTON SCIENCE LIBRARY
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Production planning and contro main
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